update pathops to circle sort

src/pathops/SkOpSegment.cpp

 /*
  * Copyright 2012 Google Inc.
  *
  * Use of this source code is governed by a BSD-style license that can be
  * found in the LICENSE file.
  */
 #include "SkIntersections.h"
 #include "SkOpSegment.h"
 #include "SkPathWriter.h"
 #include "SkTSort.h"
 
 #define F (false)      // discard the edge
 #define T (true)       // keep the edge
 
 static const bool gUnaryActiveEdge[2][2] = {
 //  from=0  from=1
 //  to=0,1  to=0,1
     {F, T}, {T, F},
 };
 
 static const bool gActiveEdge[kXOR_PathOp + 1][2][2][2][2] = {
 //                 miFrom=0                              miFrom=1
-//         miTo=0            miTo=1              miTo=0             miTo=1
-//    suFrom=0    1     suFrom=0     1      suFrom=0    1      suFrom=0    1
+//         miTo=0             miTo=1             miTo=0             miTo=1
+//     suFrom=0    1      suFrom=0    1      suFrom=0    1      suFrom=0    1
 //   suTo=0,1 suTo=0,1  suTo=0,1 suTo=0,1  suTo=0,1 suTo=0,1  suTo=0,1 suTo=0,1
     {{{{F, F}, {F, F}}, {{T, F}, {T, F}}}, {{{T, T}, {F, F}}, {{F, T}, {T, F}}}},  // mi - su
     {{{{F, F}, {F, F}}, {{F, T}, {F, T}}}, {{{F, F}, {T, T}}, {{F, T}, {T, F}}}},  // mi & su
     {{{{F, T}, {T, F}}, {{T, T}, {F, F}}}, {{{T, F}, {T, F}}, {{F, F}, {F, F}}}},  // mi | su
     {{{{F, T}, {T, F}}, {{T, F}, {F, T}}}, {{{T, F}, {F, T}}, {{F, T}, {T, F}}}},  // mi ^ su
 };
 
 #undef F
 #undef T
 
 enum {
     kOutsideTrackedTCount = 16,  // FIXME: determine what this should be
     kMissingSpanCount = 4,  // FIXME: determine what this should be
 };
 
-// note that this follows the same logic flow as build angles
-bool SkOpSegment::activeAngle(int index, int* done, SkTArray<SkOpAngle, true>* angles) {
-    if (activeAngleInner(index, done, angles)) {
-        return true;
+const SkOpAngle* SkOpSegment::activeAngle(int index, int* start, int* end, bool* done,
+        bool* sortable) const {
+    if (const SkOpAngle* result = activeAngleInner(index, start, end, done, sortable)) {
+        return result;
     }
     double referenceT = fTs[index].fT;
     int lesser = index;
     while (--lesser >= 0
             && (precisely_negative(referenceT - fTs[lesser].fT) || fTs[lesser].fTiny)) {
-        if (activeAngleOther(lesser, done, angles)) {
-            return true;
+        if (const SkOpAngle* result = activeAngleOther(lesser, start, end, done, sortable)) {
+            return result;
         }
     }
     do {
-        if (activeAngleOther(index, done, angles)) {
-            return true;
+        if (const SkOpAngle* result = activeAngleOther(index, start, end, done, sortable)) {
+            return result;
         }
         if (++index == fTs.count()) {
             break;
         }
         if (fTs[index - 1].fTiny) {
             referenceT = fTs[index].fT;
             continue;
         }
     } while (precisely_negative(fTs[index].fT - referenceT));
-    return false;
+    return NULL;
 }
 
-bool SkOpSegment::activeAngleOther(int index, int* done, SkTArray<SkOpAngle, true>* angles) {
-    SkOpSpan* span = &fTs[index];
-    SkOpSegment* other = span->fOther;
-    int oIndex = span->fOtherIndex;
-    return other->activeAngleInner(oIndex, done, angles);
-}
-
-bool SkOpSegment::activeAngleInner(int index, int* done, SkTArray<SkOpAngle, true>* angles) {
+const SkOpAngle* SkOpSegment::activeAngleInner(int index, int* start, int* end, bool* done,
+        bool* sortable) const {
     int next = nextExactSpan(index, 1);
     if (next > 0) {
-        SkOpSpan& upSpan = fTs[index];
+        const SkOpSpan& upSpan = fTs[index];
+        if (upSpan.fUnsortableStart) {
+            *sortable = false;
+            return NULL;
+        }
         if (upSpan.fWindValue || upSpan.fOppValue) {
-            addAngle(angles, index, next);
-            if (upSpan.fDone || upSpan.fUnsortableEnd) {
-                (*done)++;
-            } else if (upSpan.fWindSum != SK_MinS32) {
-                return true;
+            if (*end < 0) {
+                *start = index;
+                *end = next;
             }
-        } else if (!upSpan.fDone) {
-            upSpan.fDone = true;
-            fDoneSpans++;
+            if (!upSpan.fDone && !upSpan.fUnsortableEnd) {
+                if (upSpan.fWindSum != SK_MinS32) {
+                    return spanToAngle(index, next);
+                }
+                *done = false;
+            }
+        } else {
+            SkASSERT(upSpan.fDone);
         }
     }
     int prev = nextExactSpan(index, -1);
     // edge leading into junction
     if (prev >= 0) {
-        SkOpSpan& downSpan = fTs[prev];
+        const SkOpSpan& downSpan = fTs[prev];
+        if (downSpan.fUnsortableEnd) {
+            *sortable = false;
+            return NULL;
+        }
         if (downSpan.fWindValue || downSpan.fOppValue) {
-            addAngle(angles, index, prev);
-            if (downSpan.fDone) {
-                (*done)++;
-             } else if (downSpan.fWindSum != SK_MinS32) {
-                return true;
+            if (*end < 0) {
+                *start = index;
+                *end = prev;
             }
-        } else if (!downSpan.fDone) {
-            downSpan.fDone = true;
-            fDoneSpans++;
+            if (!downSpan.fDone) {
+                if (downSpan.fWindSum != SK_MinS32) {
+                    return spanToAngle(index, prev);
+                }
+                *done = false;
+            }
+        } else {
+            SkASSERT(downSpan.fDone);
         }
     }
-    return false;
+    return NULL;
+}
+
+const SkOpAngle* SkOpSegment::activeAngleOther(int index, int* start, int* end, bool* done,
+        bool* sortable) const {
+    const SkOpSpan* span = &fTs[index];
+    SkOpSegment* other = span->fOther;
+    int oIndex = span->fOtherIndex;
+    return other->activeAngleInner(oIndex, start, end, done, sortable);
 }
 
 SkPoint SkOpSegment::activeLeftTop(bool onlySortable, int* firstT) const {
     SkASSERT(!done());
     SkPoint topPt = {SK_ScalarMax, SK_ScalarMax};
     int count = fTs.count();
     // see if either end is not done since we want smaller Y of the pair
     bool lastDone = true;
     bool lastUnsortable = false;
     double lastT = -1;
@@ skipping 34 matching lines @@
     }
     return topPt;
 }
 
 bool SkOpSegment::activeOp(int index, int endIndex, int xorMiMask, int xorSuMask, SkPathOp op) {
     int sumMiWinding = updateWinding(endIndex, index);
     int sumSuWinding = updateOppWinding(endIndex, index);
     if (fOperand) {
         SkTSwap<int>(sumMiWinding, sumSuWinding);
     }
-    int maxWinding, sumWinding, oppMaxWinding, oppSumWinding;
-    return activeOp(xorMiMask, xorSuMask, index, endIndex, op, &sumMiWinding, &sumSuWinding,
-            &maxWinding, &sumWinding, &oppMaxWinding, &oppSumWinding);
+    return activeOp(xorMiMask, xorSuMask, index, endIndex, op, &sumMiWinding, &sumSuWinding);
 }
 
 bool SkOpSegment::activeOp(int xorMiMask, int xorSuMask, int index, int endIndex, SkPathOp op,
-        int* sumMiWinding, int* sumSuWinding,
-        int* maxWinding, int* sumWinding, int* oppMaxWinding, int* oppSumWinding) {
+        int* sumMiWinding, int* sumSuWinding) {
+    int maxWinding, sumWinding, oppMaxWinding, oppSumWinding;
     setUpWindings(index, endIndex, sumMiWinding, sumSuWinding,
-            maxWinding, sumWinding, oppMaxWinding, oppSumWinding);
+            &maxWinding, &sumWinding, &oppMaxWinding, &oppSumWinding);
     bool miFrom;
     bool miTo;
     bool suFrom;
     bool suTo;
     if (operand()) {
-        miFrom = (*oppMaxWinding & xorMiMask) != 0;
-        miTo = (*oppSumWinding & xorMiMask) != 0;
-        suFrom = (*maxWinding & xorSuMask) != 0;
-        suTo = (*sumWinding & xorSuMask) != 0;
+        miFrom = (oppMaxWinding & xorMiMask) != 0;
+        miTo = (oppSumWinding & xorMiMask) != 0;
+        suFrom = (maxWinding & xorSuMask) != 0;
+        suTo = (sumWinding & xorSuMask) != 0;
     } else {
-        miFrom = (*maxWinding & xorMiMask) != 0;
-        miTo = (*sumWinding & xorMiMask) != 0;
-        suFrom = (*oppMaxWinding & xorSuMask) != 0;
-        suTo = (*oppSumWinding & xorSuMask) != 0;
+        miFrom = (maxWinding & xorMiMask) != 0;
+        miTo = (sumWinding & xorMiMask) != 0;
+        suFrom = (oppMaxWinding & xorSuMask) != 0;
+        suTo = (oppSumWinding & xorSuMask) != 0;
     }
     bool result = gActiveEdge[op][miFrom][miTo][suFrom][suTo];
 #if DEBUG_ACTIVE_OP
     SkDebugf("%s op=%s miFrom=%d miTo=%d suFrom=%d suTo=%d result=%d\n", __FUNCTION__,
             SkPathOpsDebug::kPathOpStr[op], miFrom, miTo, suFrom, suTo, result);
 #endif
     return result;
 }
 
 bool SkOpSegment::activeWinding(int index, int endIndex) {
     int sumWinding = updateWinding(endIndex, index);
-    int maxWinding;
-    return activeWinding(index, endIndex, &maxWinding, &sumWinding);
+    return activeWinding(index, endIndex, &sumWinding);
 }
 
-bool SkOpSegment::activeWinding(int index, int endIndex, int* maxWinding, int* sumWinding) {
-    setUpWinding(index, endIndex, maxWinding, sumWinding);
-    bool from = *maxWinding != 0;
+bool SkOpSegment::activeWinding(int index, int endIndex, int* sumWinding) {
+    int maxWinding;
+    setUpWinding(index, endIndex, &maxWinding, sumWinding);
+    bool from = maxWinding != 0;
     bool to = *sumWinding  != 0;
     bool result = gUnaryActiveEdge[from][to];
     return result;
 }
 
-void SkOpSegment::addAngle(SkTArray<SkOpAngle, true>* anglesPtr, int start, int end) const {
-    SkASSERT(start != end);
-    SkOpAngle& angle = anglesPtr->push_back();
-    angle.set(this, start, end);
-}
-
 void SkOpSegment::addCancelOutsides(const SkPoint& startPt, const SkPoint& endPt,
         SkOpSegment* other) {
     int tIndex = -1;
     int tCount = fTs.count();
     int oIndex = -1;
     int oCount = other->fTs.count();
     do {
         ++tIndex;
     } while (startPt != fTs[tIndex].fPt && tIndex < tCount);
     int tIndexStart = tIndex;
@@ skipping 67 matching lines @@
 void SkOpSegment::addCoinOutsides(const SkPoint& startPt, const SkPoint& endPt,
         SkOpSegment* other) {
     // walk this to startPt
     // walk other to startPt
     // if either is > 0, add a pointer to the other, copying adjacent winding
     int tIndex = -1;
     int oIndex = -1;
     do {
         ++tIndex;
     } while (startPt != fTs[tIndex].fPt);
+    int ttIndex = tIndex;
+    bool checkOtherTMatch = false;
+    do {
+        const SkOpSpan& span = fTs[ttIndex];
+        if (startPt != span.fPt) {
+            break;
+        } 
+        if (span.fOther == other && span.fPt == startPt) {
+            checkOtherTMatch = true;
+            break;
+        }
+    } while (++ttIndex < count());
     do {
         ++oIndex;
     } while (startPt != other->fTs[oIndex].fPt);
-    if (tIndex > 0 || oIndex > 0 || fOperand != other->fOperand) {
+    bool skipAdd = false;
+    if (checkOtherTMatch) {
+        int ooIndex = oIndex;
+        do {
+            const SkOpSpan& oSpan = other->fTs[ooIndex];
+            if (startPt != oSpan.fPt) {
+                break;
+            } 
+            if (oSpan.fT == fTs[ttIndex].fOtherT) {
+                skipAdd = true;
+                break;
+            }
+        } while (++ooIndex < other->count());
+    }
+    if ((tIndex > 0 || oIndex > 0 || fOperand != other->fOperand) && !skipAdd) {
         addTPair(fTs[tIndex].fT, other, other->fTs[oIndex].fT, false, startPt);
     }
     SkPoint nextPt = startPt;
     do {
         const SkPoint* workPt;
         do {
             workPt = &fTs[++tIndex].fPt;
         } while (nextPt == *workPt);
         do {
             workPt = &other->fTs[++oIndex].fPt;
@@ skipping 55 matching lines @@
                     path->cubicTo(ePtr[1], ePtr[2], ePtr[3]);
                     break;
                 default:
                     SkASSERT(0);
             }
         }
     }
   //  return ePtr[SkPathOpsVerbToPoints(fVerb)];
 }
 
+void SkOpSegment::addEndSpan(int endIndex) {
+    int spanCount = fTs.count();
+    int angleIndex = fAngles.count();
+    int startIndex = endIndex - 1;
+    while (fTs[startIndex].fT == 1 || fTs[startIndex].fTiny) {
+        ++startIndex;
+        SkASSERT(startIndex < spanCount - 1);
+        ++endIndex;
+    }
+    fAngles.push_back().set(this, spanCount - 1, startIndex);
+#if DEBUG_ANGLE
+    fAngles.back().setID(angleIndex);
+    debugCheckPointsEqualish(endIndex, spanCount);
+#endif
+    setFromAngleIndex(endIndex, angleIndex);
+}
+
+void SkOpSegment::setFromAngleIndex(int endIndex, int angleIndex) {
+    int spanCount = fTs.count();
+    do {
+        fTs[endIndex].fFromAngleIndex = angleIndex;
+    } while (++endIndex < spanCount);
+}
+
 void SkOpSegment::addLine(const SkPoint pts[2], bool operand, bool evenOdd) {
     init(pts, SkPath::kLine_Verb, operand, evenOdd);
     fBounds.set(pts, 2);
 }
 
 // add 2 to edge or out of range values to get T extremes
 void SkOpSegment::addOtherT(int index, double otherT, int otherIndex) {
     SkOpSpan& span = fTs[index];
     if (precisely_zero(otherT)) {
         otherT = 0;
     } else if (precisely_equal(otherT, 1)) {
         otherT = 1;
     }
     span.fOtherT = otherT;
     span.fOtherIndex = otherIndex;
 }
 
 void SkOpSegment::addQuad(const SkPoint pts[3], bool operand, bool evenOdd) {
     init(pts, SkPath::kQuad_Verb, operand, evenOdd);
     fBounds.setQuadBounds(pts);
 }
 
+int SkOpSegment::addSingletonAngleDown(int endIndex, SkOpSegment** otherPtr) {
+    int startIndex = findEndSpan(endIndex);
+    SkASSERT(startIndex > 0);
+    int spanIndex = endIndex - 1;
+    fSingletonAngles.push_back().set(this, spanIndex, startIndex - 1);
+    setFromAngleIndex(startIndex, fAngles.count() + fSingletonAngles.count() - 1);
+    SkOpSegment* other;
+    do {
+        other = fTs[spanIndex].fOther;
+        if (other->fTs[0].fWindValue) {
+            break;
+        }
+        --spanIndex;
+        SkASSERT(fTs[spanIndex].fT == 1);
+    } while (true);
+    int oEndIndex = other->findStartSpan(0);
+    SkASSERT(oEndIndex > 0);
+    int otherIndex = other->fSingletonAngles.count();
+    other->fSingletonAngles.push_back().set(other, 0, oEndIndex);
+    other->setToAngleIndex(oEndIndex, other->fAngles.count() + otherIndex);
+    *otherPtr = other;
+    return otherIndex;
+}
+
+int SkOpSegment::addSingletonAngleUp(int start, SkOpSegment** otherPtr) {
+    int endIndex = findStartSpan(start);
+    SkASSERT(endIndex > 0);
+    int thisIndex = fSingletonAngles.count();
+    fSingletonAngles.push_back().set(this, start, endIndex);
+    setToAngleIndex(endIndex, fAngles.count() + thisIndex);
+    int spanIndex = start;
+    SkOpSegment* other;
+    int oCount, oStartIndex;
+    do {
+        other = fTs[spanIndex].fOther;
+        oCount = other->count();
+        oStartIndex = other->findEndSpan(oCount);
+        SkASSERT(oStartIndex > 0);
+        if (other->fTs[oStartIndex - 1].fWindValue) {
+            break;
+        }
+        ++spanIndex;
+        SkASSERT(fTs[spanIndex].fT == 0);
+    } while (true);
+    int otherIndex = other->fSingletonAngles.count();
+    other->fSingletonAngles.push_back().set(other, oCount - 1, oStartIndex - 1);
+    other->setFromAngleIndex(oStartIndex, other->fAngles.count() + otherIndex);
+    *otherPtr = other;
+    return otherIndex;
+}
+
+SkOpAngle* SkOpSegment::addSingletonAngles(int start, int step) {
+    int thisIndex = fSingletonAngles.count();
+    SkOpSegment* other;
+    int otherIndex;
+    if (step > 0) {
+        otherIndex = addSingletonAngleUp(start, &other);
+    } else {
+        otherIndex = addSingletonAngleDown(start + 1, &other);
+    }
+    fSingletonAngles[thisIndex].insert(&other->fSingletonAngles[otherIndex]);
+#if DEBUG_ANGLE
+    fSingletonAngles[thisIndex].setID(fAngles.count() + thisIndex);
+    other->fSingletonAngles[otherIndex].setID(other->fAngles.count() + otherIndex);
+#endif
+    return &fSingletonAngles[thisIndex];
+}
+
+void SkOpSegment::addStartSpan(int endIndex) {
+    int angleIndex = fAngles.count();
+    int index = 0;
+    fAngles.push_back().set(this, index, endIndex);
+#if DEBUG_ANGLE
+    fAngles.back().setID(angleIndex);
+    debugCheckPointsEqualish(index, endIndex);
+#endif
+    setToAngleIndex(endIndex, angleIndex);
+}
+
+void SkOpSegment::setToAngleIndex(int endIndex, int angleIndex) {
+    int index = 0;
+    do {
+        fTs[index].fToAngleIndex = angleIndex;
+    } while (++index < endIndex);
+}
+
     // Defer all coincident edge processing until
     // after normal intersections have been computed
 
 // no need to be tricky; insert in normal T order
 // resolve overlapping ts when considering coincidence later
 
     // add non-coincident intersection. Resulting edges are sorted in T.
 int SkOpSegment::addT(SkOpSegment* other, const SkPoint& pt, double newT) {
+    SkASSERT(this != other || fVerb == SkPath::kCubic_Verb);
+ #if 0  // this needs an even rougher association to be useful
+    SkASSERT(SkDPoint::RoughlyEqual(ptAtT(newT), pt));
+ #endif
     if (precisely_zero(newT)) {
         newT = 0;
     } else if (precisely_equal(newT, 1)) {
         newT = 1;
     }
     // FIXME: in the pathological case where there is a ton of intercepts,
     //  binary search?
     int insertedAt = -1;
-    size_t tCount = fTs.count();
+    int tCount = fTs.count();
     const SkPoint& firstPt = fPts[0];
     const SkPoint& lastPt = fPts[SkPathOpsVerbToPoints(fVerb)];
-    for (size_t index = 0; index < tCount; ++index) {
+    for (int index = 0; index < tCount; ++index) {
         // OPTIMIZATION: if there are three or more identical Ts, then
         // the fourth and following could be further insertion-sorted so
         // that all the edges are clockwise or counterclockwise.
         // This could later limit segment tests to the two adjacent
         // neighbors, although it doesn't help with determining which
         // circular direction to go in.
         const SkOpSpan& span = fTs[index];
         if (newT < span.fT) {
             insertedAt = index;
             break;
@@ skipping 17 matching lines @@
         span = fTs.append();
     }
     span->fT = newT;
     span->fOther = other;
     span->fPt = pt;
 #if 0
     // cubics, for instance, may not be exact enough to satisfy this check (e.g., cubicOp69d)
     SkASSERT(approximately_equal(xyAtT(newT).fX, pt.fX)
             && approximately_equal(xyAtT(newT).fY, pt.fY));
 #endif
+    span->fFromAngleIndex = -1;
+    span->fToAngleIndex = -1;
     span->fWindSum = SK_MinS32;
     span->fOppSum = SK_MinS32;
     span->fWindValue = 1;
     span->fOppValue = 0;
-    span->fSmall = false;
-    span->fTiny = false;
-    span->fLoop = false;
+    span->fChased = false;
     if ((span->fDone = newT == 1)) {
         ++fDoneSpans;
     }
+    span->fLoop = false;
+    span->fSmall = false;
+    span->fTiny = false;
     span->fUnsortableStart = false;
     span->fUnsortableEnd = false;
     int less = -1;
-    while (&span[less + 1] - fTs.begin() > 0 && AlmostEqualUlps(span[less].fPt, span->fPt)) {
-        if (span[less].fDone) {
-            break;
-        }
-        double tInterval = newT - span[less].fT;
-        if (precisely_negative(tInterval)) {
-            break;
-        }
+// find range of spans with nearly the same point as this one
+    while (&span[less + 1] - fTs.begin() > 0 && AlmostEqualUlps(span[less].fPt, pt)) {
         if (fVerb == SkPath::kCubic_Verb) {
+            double tInterval = newT - span[less].fT;
             double tMid = newT - tInterval / 2;
             SkDPoint midPt = dcubic_xy_at_t(fPts, tMid);
             if (!midPt.approximatelyEqual(xyAtT(span))) {
                 break;
             }
         }
-        span[less].fSmall = true;
-        bool tiny = span[less].fPt == span->fPt;
-        span[less].fTiny = tiny;
-        span[less].fDone = true;
-        if (approximately_negative(newT - span[less].fT) && tiny) {
-            if (approximately_greater_than_one(newT)) {
-                SkASSERT(&span[less] - fTs.begin() < fTs.count());
-                span[less].fUnsortableStart = true;
-                if (&span[less - 1] - fTs.begin() >= 0) {
-                    span[less - 1].fUnsortableEnd = true;
-                }
-            }
-            if (approximately_less_than_zero(span[less].fT)) {
-                SkASSERT(&span[less + 1] - fTs.begin() < fTs.count());
-                SkASSERT(&span[less] - fTs.begin() >= 0);
-                span[less + 1].fUnsortableStart = true;
-                span[less].fUnsortableEnd = true;
-            }
-        }
-        ++fDoneSpans;
         --less;
     }
     int more = 1;
-    while (fTs.end() - &span[more - 1] > 1 && AlmostEqualUlps(span[more].fPt, span->fPt)) {
-        if (span[more - 1].fDone) {
-            break;
-        }
-        double tEndInterval = span[more].fT - newT;
-        if (precisely_negative(tEndInterval)) {
-            if ((span->fTiny = span[more].fTiny)) {
-                span->fDone = true;
-                ++fDoneSpans;
-            }
-            break;
-        }
+    while (fTs.end() - &span[more - 1] > 1 && AlmostEqualUlps(span[more].fPt, pt)) {
         if (fVerb == SkPath::kCubic_Verb) {
+            double tEndInterval = span[more].fT - newT;
             double tMid = newT - tEndInterval / 2;
             SkDPoint midEndPt = dcubic_xy_at_t(fPts, tMid);
             if (!midEndPt.approximatelyEqual(xyAtT(span))) {
                 break;
             }
         }
-        span[more - 1].fSmall = true;
-        bool tiny = span[more].fPt == span->fPt;
-        span[more - 1].fTiny = tiny;
-        span[more - 1].fDone = true;
-        if (approximately_negative(span[more].fT - newT) && tiny) {
-            if (approximately_greater_than_one(span[more].fT)) {
-                span[more + 1].fUnsortableStart = true;
-                span[more].fUnsortableEnd = true;
-            }
-            if (approximately_less_than_zero(newT)) {
-                span[more].fUnsortableStart = true;
-                span[more - 1].fUnsortableEnd = true;
-            }
-        }
-        ++fDoneSpans;
         ++more;
     }
+    ++less;
+    --more;
+    while (more - 1 > less && span[more].fPt == span[more - 1].fPt
+            && span[more].fT == span[more - 1].fT) {
+        --more;
+    }
+    if (less == more) {
+        return insertedAt;
+    }
+    if (precisely_negative(span[more].fT - span[less].fT)) {
+        return insertedAt;
+    }
+// if the total range of t values is big enough, mark all tiny
+    bool tiny = span[less].fPt == span[more].fPt;
+    int index = less;
+    do {
+        fSmall = span[index].fSmall = true;
+        fTiny |= span[index].fTiny = tiny;
+        if (!span[index].fDone) {
+            span[index].fDone = true;
+            ++fDoneSpans;
+        }
+    } while (++index < more);
     return insertedAt;
 }
 
 // set spans from start to end to decrement by one
 // note this walks other backwards
 // FIXME: there's probably an edge case that can be constructed where
 // two span in one segment are separated by float epsilon on one span but
 // not the other, if one segment is very small. For this
 // case the counts asserted below may or may not be enough to separate the
 // spans. Even if the counts work out, what if the spans aren't correctly
 // sorted? It feels better in such a case to match the span's other span
 // pointer since both coincident segments must contain the same spans.
 // FIXME? It seems that decrementing by one will fail for complex paths that
 // have three or more coincident edges. Shouldn't this subtract the difference
 // between the winding values?
 /*                                      |-->                           |-->
 this     0>>>>1>>>>2>>>>3>>>4      0>>>>1>>>>2>>>>3>>>4      0>>>>1>>>>2>>>>3>>>4
 other         2<<<<1<<<<0               2<<<<1<<<<0               2<<<<1<<<<0
               ^         ^                 <--|                           <--|
            startPt    endPt        test/oTest first pos      test/oTest final pos
 */
 void SkOpSegment::addTCancel(const SkPoint& startPt, const SkPoint& endPt, SkOpSegment* other) {
     bool binary = fOperand != other->fOperand;
     int index = 0;
     while (startPt != fTs[index].fPt) {
         SkASSERT(index < fTs.count());
         ++index;
     }
-    while (index > 0 && fTs[index].fT == fTs[index - 1].fT) {
+    while (index > 0 && precisely_equal(fTs[index].fT, fTs[index - 1].fT)) {
         --index;
     }
     int oIndex = other->fTs.count();
     while (startPt != other->fTs[--oIndex].fPt) {  // look for startPt match
         SkASSERT(oIndex > 0);
     }
     double oStartT = other->fTs[oIndex].fT;
     // look for first point beyond match
-    while (startPt == other->fTs[--oIndex].fPt || oStartT == other->fTs[oIndex].fT) {
+    while (startPt == other->fTs[--oIndex].fPt || precisely_equal(oStartT, other->fTs[oIndex].fT)) {
         SkASSERT(oIndex > 0);
     }
     SkOpSpan* test = &fTs[index];
     SkOpSpan* oTest = &other->fTs[oIndex];
     SkSTArray<kOutsideTrackedTCount, SkPoint, true> outsidePts;
     SkSTArray<kOutsideTrackedTCount, SkPoint, true> oOutsidePts;
     do {
         SkASSERT(test->fT < 1);
         SkASSERT(oTest->fT < 1);
         bool decrement = test->fWindValue && oTest->fWindValue;
         bool track = test->fWindValue || oTest->fWindValue;
         bool bigger = test->fWindValue >= oTest->fWindValue;
         const SkPoint& testPt = test->fPt;
         double testT = test->fT;
         const SkPoint& oTestPt = oTest->fPt;
         double oTestT = oTest->fT;
         do {
             if (decrement) {
                 if (binary && bigger) {
                     test->fOppValue--;
                 } else {
                     decrementSpan(test);
                 }
             } else if (track) {
                 TrackOutsidePair(&outsidePts, testPt, oTestPt);
             }
             SkASSERT(index < fTs.count() - 1);
             test = &fTs[++index];
-        } while (testPt == test->fPt || testT == test->fT);
+        } while (testPt == test->fPt || precisely_equal(testT, test->fT));
         SkDEBUGCODE(int originalWindValue = oTest->fWindValue);
         do {
             SkASSERT(oTest->fT < 1);
             SkASSERT(originalWindValue == oTest->fWindValue);
             if (decrement) {
                 if (binary && !bigger) {
                     oTest->fOppValue--;
                 } else {
                     other->decrementSpan(oTest);
                 }
             } else if (track) {
                 TrackOutsidePair(&oOutsidePts, oTestPt, testPt);
             }
             if (!oIndex) {
                 break;
             }
             oTest = &other->fTs[--oIndex];
-        } while (oTestPt == oTest->fPt || oTestT == oTest->fT);
+        } while (oTestPt == oTest->fPt || precisely_equal(oTestT, oTest->fT));
     } while (endPt != test->fPt && test->fT < 1);
     // FIXME: determine if canceled edges need outside ts added
     int outCount = outsidePts.count();
     if (!done() && outCount) {
         addCancelOutsides(outsidePts[0], outsidePts[1], other);
         if (outCount > 2) {
             addCancelOutsides(outsidePts[outCount - 2], outsidePts[outCount - 1], other);
         }
     }
     if (!other->done() && oOutsidePts.count()) {
         other->addCancelOutsides(oOutsidePts[0], oOutsidePts[1], this);
     }
 }
 
-int SkOpSegment::addSelfT(SkOpSegment* other, const SkPoint& pt, double newT) {
+int SkOpSegment::addSelfT(const SkPoint& pt, double newT) {
     // if the tail nearly intersects itself but not quite, the caller records this separately
-    int result = addT(other, pt, newT);
+    int result = addT(this, pt, newT);
     SkOpSpan* span = &fTs[result];
-    span->fLoop = true;
+    fLoop = span->fLoop = true;
     return result;
 }
 
+void SkOpSegment::addSimpleAngle(int index) {
+    if (index == 0) {
+        SkASSERT(!fTs[index].fTiny && fTs[index + 1].fT > 0);
+        addStartSpan(1);
+    } else {
+        SkASSERT(!fTs[index - 1].fTiny && fTs[index - 1].fT < 1);
+        addEndSpan(index);
+    }
+    SkOpSpan& span = fTs[index];
+    SkOpSegment* other = span.fOther;
+    SkOpSpan& oSpan = other->fTs[span.fOtherIndex];
+    SkOpAngle* angle, * oAngle;
+    if (index == 0) {
+        SkASSERT(span.fOtherIndex - 1 >= 0);
+        SkASSERT(span.fOtherT == 1);
+        SkDEBUGCODE(SkOpSpan& oPrior = other->fTs[span.fOtherIndex - 1]);
+        SkASSERT(!oPrior.fTiny && oPrior.fT < 1);
+        other->addEndSpan(span.fOtherIndex);
+        angle = &this->angle(span.fToAngleIndex);
+        oAngle = &other->angle(oSpan.fFromAngleIndex);
+    } else {
+        SkASSERT(span.fOtherIndex + 1 < other->count());
+        SkASSERT(span.fOtherT == 0);
+        SkASSERT(!oSpan.fTiny && (other->fTs[span.fOtherIndex + 1].fT > 0
+                || (other->fTs[span.fOtherIndex + 1].fFromAngleIndex < 0
+                && other->fTs[span.fOtherIndex + 1].fToAngleIndex < 0)));
+        int oIndex = 1;
+        do {
+            const SkOpSpan& osSpan = other->span(oIndex);
+            if (osSpan.fFromAngleIndex >= 0 || osSpan.fT > 0) {
+                break;
+            }
+            ++oIndex;
+            SkASSERT(oIndex < other->count());
+        } while (true);
+        other->addStartSpan(oIndex);
+        angle = &this->angle(span.fFromAngleIndex);
+        oAngle = &other->angle(oSpan.fToAngleIndex);
+    }
+    angle->insert(oAngle);
+}
+
+bool SkOpSegment::alignSpan(int index, double thisT, const SkPoint& thisPt) {
+    bool aligned = false;
+    SkOpSpan* span = &fTs[index];
+    SkOpSegment* other = span->fOther;
+    int oIndex = span->fOtherIndex;
+    SkOpSpan* oSpan = &other->fTs[oIndex];
+    if (span->fT != thisT) {
+        span->fT = thisT;
+        oSpan->fOtherT = thisT;
+        aligned = true;
+    }
+    if (span->fPt != thisPt) {
+        span->fPt = thisPt;
+        oSpan->fPt = thisPt;
+        aligned = true;
+    }
+    double oT = oSpan->fT;
+    if (oT == 0 || oT == 1) {
+        return aligned;
+    }
+    int oStart = other->nextSpan(oIndex, -1) + 1;
+    int oEnd = other->nextSpan(oIndex, 1);
+    oSpan = &other->fTs[oStart];
+    oT = oSpan->fT;
+    bool oAligned = false;
+    if (oSpan->fPt != thisPt) {
+        oAligned |= other->alignSpan(oStart, oT, thisPt);
+    }
+    int otherIndex = oStart;
+    while (++otherIndex < oEnd) {
+        SkOpSpan* oNextSpan = &other->fTs[otherIndex];
+        if (oNextSpan->fT != oT || oNextSpan->fPt != thisPt) {
+            oAligned |= other->alignSpan(otherIndex, oT, thisPt);
+        }
+    }
+    if (oAligned) {
+        other->alignSpanState(oStart, oEnd);
+    }
+    return aligned;
+}
+
+void SkOpSegment::alignSpanState(int start, int end) {
+    SkOpSpan* lastSpan = &fTs[--end];
+    bool allSmall = lastSpan->fSmall;
+    bool allTiny = lastSpan->fTiny;
+    bool allDone = lastSpan->fDone;
+    SkDEBUGCODE(int winding = lastSpan->fWindValue);
+    SkDEBUGCODE(int oppWinding = lastSpan->fOppValue);
+    int index = start;
+    while (index < end) {
+        SkOpSpan* span = &fTs[index];
+        span->fSmall = allSmall;
+        span->fTiny = allTiny;
+        if (span->fDone != allDone) {
+            span->fDone = allDone;
+            fDoneSpans += allDone ? 1 : -1;
+        }
+        SkASSERT(span->fWindValue == winding);
+        SkASSERT(span->fOppValue == oppWinding);
+        ++index;
+    }
+}
+
 void SkOpSegment::bumpCoincidentThis(const SkOpSpan& oTest, bool binary, int* indexPtr,
         SkTArray<SkPoint, true>* outsideTs) {
     int index = *indexPtr;
     int oWindValue = oTest.fWindValue;
     int oOppValue = oTest.fOppValue;
     if (binary) {
         SkTSwap<int>(oWindValue, oOppValue);
     }
     SkOpSpan* const test = &fTs[index];
     SkOpSpan* end = test;
     const SkPoint& oStartPt = oTest.fPt;
     do {
         if (bumpSpan(end, oWindValue, oOppValue)) {
             TrackOutside(outsideTs, oStartPt);
         }
         end = &fTs[++index];
-    } while ((end->fPt == test->fPt || end->fT == test->fT) && end->fT < 1);
+    } while ((end->fPt == test->fPt || precisely_equal(end->fT, test->fT)) && end->fT < 1);
     *indexPtr = index;
 }
 
 // because of the order in which coincidences are resolved, this and other
 // may not have the same intermediate points. Compute the corresponding
 // intermediate T values (using this as the master, other as the follower)
 // and walk other conditionally -- hoping that it catches up in the end
 void SkOpSegment::bumpCoincidentOther(const SkOpSpan& test, int* oIndexPtr,
         SkTArray<SkPoint, true>* oOutsidePts) {
     int oIndex = *oIndexPtr;
     SkOpSpan* const oTest = &fTs[oIndex];
     SkOpSpan* oEnd = oTest;
-    const SkPoint& startPt = test.fPt;
     const SkPoint& oStartPt = oTest->fPt;
     double oStartT = oTest->fT;
-    if (oStartPt == oEnd->fPt || oStartT == oEnd->fT) {
+#if 0  // FIXME : figure out what disabling this breaks
+    const SkPoint& startPt = test.fPt;
+    // this is always true since oEnd == oTest && oStartPt == oTest->fPt -- find proper condition
+    if (oStartPt == oEnd->fPt || precisely_equal(oStartT, oEnd->fT)) {
         TrackOutside(oOutsidePts, startPt);
     }
-    while (oStartPt == oEnd->fPt || oStartT == oEnd->fT) {
+#endif
+    while (oStartPt == oEnd->fPt || precisely_equal(oStartT, oEnd->fT)) {
         zeroSpan(oEnd);
         oEnd = &fTs[++oIndex];
     }
     *oIndexPtr = oIndex;
 }
 
 // FIXME: need to test this case:
 // contourA has two segments that are coincident
 // contourB has two segments that are coincident in the same place
 // each ends up with +2/0 pairs for winding count
 // since logic below doesn't transfer count (only increments/decrements) can this be
 // resolved to +4/0 ?
 
 // set spans from start to end to increment the greater by one and decrement
 // the lesser
 void SkOpSegment::addTCoincident(const SkPoint& startPt, const SkPoint& endPt, double endT,
         SkOpSegment* other) {
     bool binary = fOperand != other->fOperand;
     int index = 0;
     while (startPt != fTs[index].fPt) {
         SkASSERT(index < fTs.count());
         ++index;
     }
     double startT = fTs[index].fT;
-    while (index > 0 && fTs[index - 1].fT == startT) {
+    while (index > 0 && precisely_equal(fTs[index - 1].fT, startT)) {
         --index;
     }
     int oIndex = 0;
     while (startPt != other->fTs[oIndex].fPt) {
         SkASSERT(oIndex < other->fTs.count());
         ++oIndex;
     }
     double oStartT = other->fTs[oIndex].fT;
-    while (oIndex > 0 && other->fTs[oIndex - 1].fT == oStartT) {
+    while (oIndex > 0 && precisely_equal(other->fTs[oIndex - 1].fT, oStartT)) {
         --oIndex;
     }
     SkSTArray<kOutsideTrackedTCount, SkPoint, true> outsidePts;
     SkSTArray<kOutsideTrackedTCount, SkPoint, true> oOutsidePts;
     SkOpSpan* test = &fTs[index];
     const SkPoint* testPt = &test->fPt;
     double testT = test->fT;
     SkOpSpan* oTest = &other->fTs[oIndex];
     const SkPoint* oTestPt = &oTest->fPt;
     SkASSERT(AlmostEqualUlps(*testPt, *oTestPt));
     do {
         SkASSERT(test->fT < 1);
         SkASSERT(oTest->fT < 1);
-#if 0
-        if (test->fDone || oTest->fDone) {
-#else  // consolidate the winding count even if done
+
+        // consolidate the winding count even if done
         if ((test->fWindValue == 0 && test->fOppValue == 0)
                 || (oTest->fWindValue == 0 && oTest->fOppValue == 0)) {
-#endif
             SkDEBUGCODE(int firstWind = test->fWindValue);
             SkDEBUGCODE(int firstOpp = test->fOppValue);
             do {
                 SkASSERT(firstWind == fTs[index].fWindValue);
                 SkASSERT(firstOpp == fTs[index].fOppValue);
                 ++index;
                 SkASSERT(index < fTs.count());
             } while (*testPt == fTs[index].fPt);
             SkDEBUGCODE(firstWind = oTest->fWindValue);
             SkDEBUGCODE(firstOpp = oTest->fOppValue);
             do {
                 SkASSERT(firstWind == other->fTs[oIndex].fWindValue);
                 SkASSERT(firstOpp == other->fTs[oIndex].fOppValue);
                 ++oIndex;
                 SkASSERT(oIndex < other->fTs.count());
             } while (*oTestPt == other->fTs[oIndex].fPt);
         } else {
             if (!binary || test->fWindValue + oTest->fOppValue >= 0) {
                 bumpCoincidentThis(*oTest, binary, &index, &outsidePts);
                 other->bumpCoincidentOther(*test, &oIndex, &oOutsidePts);
             } else {
                 other->bumpCoincidentThis(*test, binary, &oIndex, &oOutsidePts);
                 bumpCoincidentOther(*oTest, &index, &outsidePts);
             }
         }
         test = &fTs[index];
         testPt = &test->fPt;
         testT = test->fT;
-        if (endPt == *testPt || endT == testT) {
+        oTest = &other->fTs[oIndex];
+        oTestPt = &oTest->fPt;
+        if (endPt == *testPt || precisely_equal(endT, testT)) {
             break;
         }
-        oTest = &other->fTs[oIndex];
-        oTestPt = &oTest->fPt;
-        SkASSERT(AlmostEqualUlps(*testPt, *oTestPt));
+//        SkASSERT(AlmostEqualUlps(*testPt, *oTestPt));
     } while (endPt != *oTestPt);
-    if (endPt != *testPt && endT != testT) {  // in rare cases, one may have ended before the other
+    // in rare cases, one may have ended before the other
+    if (endPt != *testPt && !precisely_equal(endT, testT)) {
         int lastWind = test[-1].fWindValue;
         int lastOpp = test[-1].fOppValue;
         bool zero = lastWind == 0 && lastOpp == 0;
         do {
             if (test->fWindValue || test->fOppValue) {
                 test->fWindValue = lastWind;
                 test->fOppValue = lastOpp;
                 if (zero) {
                     test->fDone = true;
                     ++fDoneSpans;
                 }
             }
             test = &fTs[++index];
             testPt = &test->fPt;
         } while (endPt != *testPt);
-   }
+    }
+    if (endPt != *oTestPt) {
+        // look ahead to see if zeroing more spans will allows us to catch up
+        int oPeekIndex = oIndex;
+        bool success = true;
+        SkOpSpan* oPeek;
+        do {
+            oPeek = &other->fTs[oPeekIndex];
+            if (oPeek->fT == 1) {
+                success = false;
+                break;
+            }
+            ++oPeekIndex;
+        } while (endPt != oPeek->fPt);
+        if (success) {
+            // make sure the matching point completes the coincidence span
+            success = false;
+            do {
+                if (oPeek->fOther == this) {
+                    success = true;
+                    break;
+                }
+                oPeek = &other->fTs[++oPeekIndex];
+            } while (endPt == oPeek->fPt);
+        }
+        if (success) {
+            do {
+                if (!binary || test->fWindValue + oTest->fOppValue >= 0) {
+                    other->bumpCoincidentOther(*test, &oIndex, &oOutsidePts);
+                } else {
+                    other->bumpCoincidentThis(*test, binary, &oIndex, &oOutsidePts);
+                }
+                oTest = &other->fTs[oIndex];
+                oTestPt = &oTest->fPt;
+            } while (endPt != *oTestPt);
+        }
+    }
     int outCount = outsidePts.count();
     if (!done() && outCount) {
         addCoinOutsides(outsidePts[0], endPt, other);
     }
     if (!other->done() && oOutsidePts.count()) {
         other->addCoinOutsides(oOutsidePts[0], endPt, this);
     }
 }
 
 // FIXME: this doesn't prevent the same span from being added twice
 // fix in caller, SkASSERT here?
-void SkOpSegment::addTPair(double t, SkOpSegment* other, double otherT, bool borrowWind,
+const SkOpSpan* SkOpSegment::addTPair(double t, SkOpSegment* other, double otherT, bool borrowWind,
                            const SkPoint& pt) {
     int tCount = fTs.count();
     for (int tIndex = 0; tIndex < tCount; ++tIndex) {
         const SkOpSpan& span = fTs[tIndex];
         if (!approximately_negative(span.fT - t)) {
             break;
         }
         if (approximately_negative(span.fT - t) && span.fOther == other
                 && approximately_equal(span.fOtherT, otherT)) {
 #if DEBUG_ADD_T_PAIR
             SkDebugf("%s addTPair duplicate this=%d %1.9g other=%d %1.9g\n",
                     __FUNCTION__, fID, t, other->fID, otherT);
 #endif
-            return;
+            return NULL;
         }
     }
 #if DEBUG_ADD_T_PAIR
     SkDebugf("%s addTPair this=%d %1.9g other=%d %1.9g\n",
             __FUNCTION__, fID, t, other->fID, otherT);
 #endif
     int insertedAt = addT(other, pt, t);
     int otherInsertedAt = other->addT(this, pt, otherT);
     addOtherT(insertedAt, otherT, otherInsertedAt);
     other->addOtherT(otherInsertedAt, t, insertedAt);
     matchWindingValue(insertedAt, t, borrowWind);
     other->matchWindingValue(otherInsertedAt, otherT, borrowWind);
+    return &span(insertedAt);
 }
 
-void SkOpSegment::addTwoAngles(int start, int end, SkTArray<SkOpAngle, true>* angles) const {
-    // add edge leading into junction
-    int min = SkMin32(end, start);
-    if (fTs[min].fWindValue > 0 || fTs[min].fOppValue != 0) {
-        addAngle(angles, end, start);
+const SkOpAngle& SkOpSegment::angle(int index) const {
+    SkASSERT(index >= 0);
+    int count = fAngles.count();
+    if (index < count) {
+        return fAngles[index];
     }
-    // add edge leading away from junction
-    int step = SkSign32(end - start);
-    int tIndex = nextExactSpan(end, step);
-    min = SkMin32(end, tIndex);
-    if (tIndex >= 0 && (fTs[min].fWindValue > 0 || fTs[min].fOppValue != 0)) {
-        addAngle(angles, end, tIndex);
-    }
+    index -= count;
+    count = fSingletonAngles.count();
+    SkASSERT(index < count);
+    return fSingletonAngles[index];
 }
 
 bool SkOpSegment::betweenPoints(double midT, const SkPoint& pt1, const SkPoint& pt2) const {
     const SkPoint midPt = ptAtT(midT);
     SkPathOpsBounds bounds;
     bounds.set(pt1.fX, pt1.fY, pt2.fX, pt2.fY);
     bounds.sort();
     return bounds.almostContains(midPt);
 }
 
 bool SkOpSegment::betweenTs(int lesser, double testT, int greater) const {
     if (lesser > greater) {
         SkTSwap<int>(lesser, greater);
     }
     return approximately_between(fTs[lesser].fT, testT, fTs[greater].fT);
 }
 
-// note that this follows the same logic flow as active angle
-bool SkOpSegment::buildAngles(int index, SkTArray<SkOpAngle, true>* angles, bool allowOpp) const {
-    double referenceT = fTs[index].fT;
-    const SkPoint& referencePt = fTs[index].fPt;
-    int lesser = index;
-    while (--lesser >= 0 && (allowOpp || fTs[lesser].fOther->fOperand == fOperand)
-            && (precisely_negative(referenceT - fTs[lesser].fT) || fTs[lesser].fTiny)) {
-        buildAnglesInner(lesser, angles);
+// in extreme cases (like the buffer overflow test) return false to abort
+// for now, if one t value represents two different points, then the values are too extreme
+// to generate meaningful results
+bool SkOpSegment::calcAngles() {
+    int spanCount = fTs.count();
+    if (spanCount <= 2) {
+        return spanCount == 2;
     }
-    do {
-        buildAnglesInner(index, angles);
-        if (++index == fTs.count()) {
-            break;
-        }
-        if (!allowOpp && fTs[index].fOther->fOperand != fOperand) {
-            break;
-        }
-        if (fTs[index - 1].fTiny) {
-            referenceT = fTs[index].fT;
-            continue;
-        }
-        if (!precisely_negative(fTs[index].fT - referenceT) && fTs[index].fPt == referencePt) {
-            // FIXME
-            // testQuad8 generates the wrong output unless false is returned here. Other tests will
-            // take this path although they aren't required to. This means that many go much slower
-            // because of this sort fail.
- //           SkDebugf("!!!\n");
+    int index = 1;
+    const SkOpSpan* firstSpan = &fTs[index];
+    int activePrior = checkSetAngle(0);
+    const SkOpSpan* span = &fTs[0];
+    if (firstSpan->fT == 0 || span->fTiny || span->fOtherT != 1 || span->fOther->multipleEnds()) {
+        index = findStartSpan(0);  // curve start intersects
+        if (index < 0) {
             return false;
         }
-    } while (precisely_negative(fTs[index].fT - referenceT));
+        if (activePrior >= 0) {
+            addStartSpan(index);
+        }
+    }
+    bool addEnd;
+    int endIndex = spanCount - 1;
+    span = &fTs[endIndex - 1];
+    if ((addEnd = span->fT == 1 || span->fTiny)) {  // if curve end intersects
+        endIndex = findEndSpan(endIndex);
+        if (endIndex < 0) {
+            return false;
+        }
+    } else {
+        addEnd = fTs[endIndex].fOtherT != 0 || fTs[endIndex].fOther->multipleStarts();
+    }
+    SkASSERT(endIndex >= index);
+    int prior = 0;
+    while (index < endIndex) {
+        const SkOpSpan& fromSpan = fTs[index];  // for each intermediate intersection
+        const SkOpSpan* lastSpan;
+        span = &fromSpan;
+        int start = index;
+        do {
+            lastSpan = span;
+            span = &fTs[++index];
+            SkASSERT(index < spanCount);
+            if (!precisely_negative(span->fT - lastSpan->fT) && !lastSpan->fTiny) {
+                break;
+            }
+            if (!SkDPoint::ApproximatelyEqual(lastSpan->fPt, span->fPt)) {
+                return false;
+            }
+        } while (true);
+        int angleIndex = fAngles.count();
+        int priorAngleIndex;
+        if (activePrior >= 0) {
+            int pActive = firstActive(prior);
+            SkASSERT(pActive < start);
+            fAngles.push_back().set(this, start, pActive);
+            priorAngleIndex = angleIndex++;
+    #if DEBUG_ANGLE
+            fAngles.back().setID(priorAngleIndex);
+    #endif
+        }
+        int active = checkSetAngle(start);
+        if (active >= 0) {
+            SkASSERT(active < index);
+            fAngles.push_back().set(this, active, index);
+    #if DEBUG_ANGLE
+            fAngles.back().setID(angleIndex);
+    #endif
+        }
+    #if DEBUG_ANGLE
+        debugCheckPointsEqualish(start, index);
+    #endif
+        prior = start;
+        do {
+            const SkOpSpan* startSpan = &fTs[start - 1];
+            if (!startSpan->fSmall || startSpan->fFromAngleIndex >= 0
+                    || startSpan->fToAngleIndex >= 0) {
+                break;
+            }
+            --start;
+        } while (start > 0);
+        do {
+            if (activePrior >= 0) {
+                SkASSERT(fTs[start].fFromAngleIndex == -1);
+                fTs[start].fFromAngleIndex = priorAngleIndex;
+            }
+            if (active >= 0) {
+                SkASSERT(fTs[start].fToAngleIndex == -1);
+                fTs[start].fToAngleIndex = angleIndex;
+            }
+        } while (++start < index);
+        activePrior = active;
+    }
+    if (addEnd && activePrior >= 0) {
+        addEndSpan(endIndex);
+    }
     return true;
 }
 
-void SkOpSegment::buildAnglesInner(int index, SkTArray<SkOpAngle, true>* angles) const {
-    const SkOpSpan* span = &fTs[index];
-    SkOpSegment* other = span->fOther;
-// if there is only one live crossing, and no coincidence, continue
-// in the same direction
-// if there is coincidence, the only choice may be to reverse direction
-    // find edge on either side of intersection
-    int oIndex = span->fOtherIndex;
-    // if done == -1, prior span has already been processed
-    int step = 1;
-    int next = other->nextExactSpan(oIndex, step);
-   if (next < 0) {
-        step = -step;
-        next = other->nextExactSpan(oIndex, step);
+int SkOpSegment::checkSetAngle(int tIndex) const {
+    const SkOpSpan* span = &fTs[tIndex];
+    while (span->fTiny /* || span->fSmall */) {
+        span = &fTs[++tIndex];
     }
-    // add candidate into and away from junction
-    other->addTwoAngles(next, oIndex, angles);
+    return isCanceled(tIndex) ? -1 : tIndex;
 }
 
-int SkOpSegment::computeSum(int startIndex, int endIndex, SkOpAngle::IncludeType includeType,
-        SkTArray<SkOpAngle, true>* angles, SkTArray<SkOpAngle*, true>* sorted) {
-    addTwoAngles(startIndex, endIndex, angles);
-    if (!buildAngles(endIndex, angles, includeType == SkOpAngle::kBinaryOpp)) {
+// at this point, the span is already ordered, or unorderable, or unsortable
+int SkOpSegment::computeSum(int startIndex, int endIndex, SkOpAngle::IncludeType includeType) {
+    SkASSERT(includeType != SkOpAngle::kUnaryXor);
+    SkOpAngle* firstAngle = spanToAngle(endIndex, startIndex);
+    if (NULL == firstAngle) {
         return SK_NaN32;
     }
-    int angleCount = angles->count();
-    // abort early before sorting if an unsortable (not an unorderable) angle is found
-    if (includeType != SkOpAngle::kUnaryXor) {
-        int firstIndex = -1;
-        while (++firstIndex < angleCount) {
-            const SkOpAngle& angle = (*angles)[firstIndex];
-            if (angle.segment()->windSum(&angle) != SK_MinS32) {
-                break;
-            }
-        }
-        if (firstIndex == angleCount) {
-            return SK_MinS32;
-        }
-    }
-    bool sortable = SortAngles2(*angles, sorted);
-#if DEBUG_SORT_RAW
-    if (sorted->count() > 0) {
-        (*sorted)[0]->segment()->debugShowSort(__FUNCTION__, *sorted, 0, 0, 0, sortable);
-    }
-#endif
-    if (!sortable) {
-        return SK_NaN32;
-    }
-    if (includeType == SkOpAngle::kUnaryXor) {
-        return SK_MinS32;
-    }
     // if all angles have a computed winding,
     //  or if no adjacent angles are orderable,
     //  or if adjacent orderable angles have no computed winding,
     //  there's nothing to do
     // if two orderable angles are adjacent, and one has winding computed, transfer to the other
-    const SkOpAngle* baseAngle = NULL;
-    int last = angleCount;
-    int finalInitialOrderable = -1;
+    SkOpAngle* baseAngle = NULL;
     bool tryReverse = false;
     // look for counterclockwise transfers
+    SkOpAngle* angle = firstAngle;
     do {
-        int index = 0;
+        int testWinding = angle->segment()->windSum(angle);
+        if (SK_MinS32 != testWinding && !angle->unorderable()) {
+            baseAngle = angle;
+            continue;
+        }
+        if (angle->unorderable()) {
+            baseAngle = NULL;
+            tryReverse = true;
+            continue;
+        }
+        if (baseAngle) {
+            ComputeOneSum(baseAngle, angle, includeType);
+            baseAngle = SK_MinS32 != angle->segment()->windSum(angle) ? angle : NULL;
+            tryReverse |= !baseAngle;
+        }
+    } while ((angle = angle->next()) != firstAngle);
+    if (baseAngle && SK_MinS32 == firstAngle->segment()->windSum(angle)) {
+        firstAngle = baseAngle;
+        tryReverse = true;
+    }
+    if (tryReverse) {
+        baseAngle = NULL;
+        angle = firstAngle;
         do {
-            SkOpAngle* testAngle = (*sorted)[index];
-            int testWinding = testAngle->segment()->windSum(testAngle);
-            if (SK_MinS32 != testWinding && !testAngle->unorderable()) {
-                baseAngle = testAngle;
+            int testWinding = angle->segment()->windSum(angle);
+            if (SK_MinS32 != testWinding) {
+                baseAngle = angle;
                 continue;
             }
-            if (testAngle->unorderable()) {
+            if (angle->unorderable()) {
                 baseAngle = NULL;
-                tryReverse = true;
                 continue;
             }
             if (baseAngle) {
-                ComputeOneSum(baseAngle, testAngle, includeType);
-                baseAngle = SK_MinS32 != testAngle->segment()->windSum(testAngle) ? testAngle
-                        : NULL;
-                tryReverse |= !baseAngle;
-                continue;
+                ComputeOneSumReverse(baseAngle, angle, includeType);
+                baseAngle = SK_MinS32 != angle->segment()->windSum(angle) ? angle : NULL;
             }
-            if (finalInitialOrderable + 1 == index) {
-                finalInitialOrderable = index;
-            }
-        } while (++index != last);
-        if (finalInitialOrderable < 0) {
-            break;
-        }
-        last = finalInitialOrderable + 1;
-        finalInitialOrderable = -2;  // make this always negative the second time through
-    } while (baseAngle);
-    if (tryReverse) {
-        baseAngle = NULL;
-        last = 0;
-        finalInitialOrderable = angleCount;
-        do {
-            int index = angleCount;
-            while (--index >= last) {
-                SkOpAngle* testAngle = (*sorted)[index];
-                int testWinding = testAngle->segment()->windSum(testAngle);
-                if (SK_MinS32 != testWinding) {
-                    baseAngle = testAngle;
-                    continue;
-                }
-                if (testAngle->unorderable()) {
-                    baseAngle = NULL;
-                    continue;
-                }
-                if (baseAngle) {
-                    ComputeOneSumReverse(baseAngle, testAngle, includeType);
-                    baseAngle = SK_MinS32 != testAngle->segment()->windSum(testAngle) ? testAngle
-                            : NULL;
-                    continue;
-                }
-                if (finalInitialOrderable - 1 == index) {
-                    finalInitialOrderable = index;
-                }
-            }
-            if (finalInitialOrderable >= angleCount) {
-                break;
-            }
-            last = finalInitialOrderable;
-            finalInitialOrderable = angleCount + 1;  // make this inactive 2nd time through
-        } while (baseAngle);
+        } while ((angle = angle->previous()) != firstAngle);
     }
     int minIndex = SkMin32(startIndex, endIndex);
     return windSum(minIndex);
 }
 
 void SkOpSegment::ComputeOneSum(const SkOpAngle* baseAngle, SkOpAngle* nextAngle,
         SkOpAngle::IncludeType includeType) {
     const SkOpSegment* baseSegment = baseAngle->segment();
     int sumMiWinding = baseSegment->updateWindingReverse(baseAngle);
     int sumSuWinding;
@@ skipping 43 matching lines @@
         last = nextSegment->markAngle(maxWinding, sumWinding, oppMaxWinding, oppSumWinding,
                 nextAngle);
     } else {
         nextSegment->setUpWindings(nextAngle->end(), nextAngle->start(), &sumMiWinding,
                 &maxWinding, &sumWinding);
         last = nextSegment->markAngle(maxWinding, sumWinding, nextAngle);
     }
     nextAngle->setLastMarked(last);
 }
 
+void SkOpSegment::constructLine(SkPoint shortLine[2]) {
+    addLine(shortLine, false, false);
+    addT(NULL, shortLine[0], 0);
+    addT(NULL, shortLine[1], 1);
+    addStartSpan(1);
+    addEndSpan(1);
+    SkOpAngle& angle = fAngles.push_back();
+    angle.set(this, 0, 1);
+}
+
 int SkOpSegment::crossedSpanY(const SkPoint& basePt, SkScalar* bestY, double* hitT,
                               bool* hitSomething, double mid, bool opp, bool current) const {
     SkScalar bottom = fBounds.fBottom;
     int bestTIndex = -1;
     if (bottom <= *bestY) {
         return bestTIndex;
     }
     SkScalar top = fBounds.fTop;
     if (top >= basePt.fY) {
         return bestTIndex;
@@ skipping 103 matching lines @@
         span->fOppValue &= 1;
     }
     if (!span->fWindValue && !span->fOppValue) {
         span->fDone = true;
         ++fDoneSpans;
         return true;
     }
     return false;
 }
 
+const SkOpSpan& SkOpSegment::firstSpan(const SkOpSpan& thisSpan) const {
+    const SkOpSpan* firstSpan = &thisSpan; // rewind to the start
+    const SkOpSpan* beginSpan = fTs.begin();
+    const SkPoint& testPt = thisSpan.fPt;
+    while (firstSpan > beginSpan && firstSpan[-1].fPt == testPt) {
+        --firstSpan;
+    }
+    return *firstSpan;
+}
+
+const SkOpSpan& SkOpSegment::lastSpan(const SkOpSpan& thisSpan) const {
+    const SkOpSpan* endSpan = fTs.end() - 1;  // last can't be small
+    const SkOpSpan* lastSpan = &thisSpan;  // find the end
+    const SkPoint& testPt = thisSpan.fPt;
+    while (lastSpan < endSpan && lastSpan[1].fPt == testPt) {
+        ++lastSpan;
+    }
+    return *lastSpan;
+}
+
+// with a loop, the comparison is move involved
+// scan backwards and forwards to count all matching points
+// (verify that there are twp scans marked as loops)
+// compare that against 2 matching scans for loop plus other results
+bool SkOpSegment::calcLoopSpanCount(const SkOpSpan& thisSpan, int* smallCounts) {
+    const SkOpSpan& firstSpan = this->firstSpan(thisSpan); // rewind to the start
+    const SkOpSpan& lastSpan = this->lastSpan(thisSpan);  // find the end
+    double firstLoopT = -1, lastLoopT = -1;
+    const SkOpSpan* testSpan = &firstSpan - 1;
+    while (++testSpan <= &lastSpan) {
+        if (testSpan->fLoop) {
+            firstLoopT = testSpan->fT;
+            break;
+        }
+    }
+    testSpan = &lastSpan + 1;
+    while (--testSpan >= &firstSpan) {
+        if (testSpan->fLoop) {
+            lastLoopT = testSpan->fT;
+            break;
+        }
+    }
+    SkASSERT((firstLoopT == -1) == (lastLoopT == -1));
+    if (firstLoopT == -1) {
+        return false;
+    }
+    SkASSERT(firstLoopT < lastLoopT);
+    testSpan = &firstSpan - 1;
+    smallCounts[0] = smallCounts[1] = 0;
+    while (++testSpan <= &lastSpan) {
+        SkASSERT(approximately_equal(testSpan->fT, firstLoopT) +
+                approximately_equal(testSpan->fT, lastLoopT) == 1);
+        smallCounts[approximately_equal(testSpan->fT, lastLoopT)]++;
+    }
+    return true;
+}
+
+// see if spans with two or more intersections have the same number on the other end
+void SkOpSegment::checkDuplicates() {
+    debugValidate();
+    SkSTArray<kMissingSpanCount, MissingSpan, true> missingSpans;
+    int index;
+    int endIndex = 0;
+    bool endFound;
+    do {
+        index = endIndex;
+        endIndex = nextExactSpan(index, 1);
+        if ((endFound = endIndex < 0)) {
+            endIndex = count();
+        }
+        int dupCount = endIndex - index;
+        if (dupCount < 2) {
+            continue;
+        }
+        do {
+            const SkOpSpan* thisSpan = &fTs[index];
+            SkOpSegment* other = thisSpan->fOther;
+            int oIndex = thisSpan->fOtherIndex;
+            int oStart = other->nextExactSpan(oIndex, -1) + 1;
+            int oEnd = other->nextExactSpan(oIndex, 1);
+            if (oEnd < 0) {
+                oEnd = other->count();
+            }
+            int oCount = oEnd - oStart;
+            // force the other to match its t and this pt if not on an end point
+            if (oCount != dupCount) {
+                MissingSpan& missing = missingSpans.push_back();
+                missing.fOther = NULL;
+                SkDEBUGCODE(sk_bzero(&missing, sizeof(missing)));
+                missing.fPt = thisSpan->fPt;
+                const SkOpSpan& oSpan = other->span(oIndex);
+                if (oCount > dupCount) {
+                    missing.fSegment = this;
+                    missing.fT = thisSpan->fT;
+                    other->checkLinks(&oSpan, &missingSpans);
+                } else {
+                    missing.fSegment = other;
+                    missing.fT = oSpan.fT;
+                    checkLinks(thisSpan, &missingSpans);
+                }
+                if (!missingSpans.back().fOther) {
+                    missingSpans.pop_back();
+                }
+            }
+        } while (++index < endIndex);
+    } while (!endFound);
+    int missingCount = missingSpans.count();
+    if (missingCount == 0) {
+        return;
+    }
+    SkSTArray<kMissingSpanCount, MissingSpan, true> missingCoincidence;
+    for (index = 0; index < missingCount; ++index)  {
+        MissingSpan& missing = missingSpans[index];
+        SkOpSegment* missingOther = missing.fOther;
+        if (missing.fSegment == missing.fOther) {
+            continue;
+        }
+        const SkOpSpan* added = missing.fSegment->addTPair(missing.fT, missingOther,
+                missing.fOtherT, false, missing.fPt);
+        if (added && added->fSmall) {
+            missing.fSegment->checkSmallCoincidence(*added, &missingCoincidence);
+        }
+    }
+    for (index = 0; index < missingCount; ++index)  {
+        MissingSpan& missing = missingSpans[index];
+        missing.fSegment->fixOtherTIndex();
+        missing.fOther->fixOtherTIndex();
+    }
+    for (index = 0; index < missingCoincidence.count(); ++index) {
+        MissingSpan& missing = missingCoincidence[index];
+        missing.fSegment->fixOtherTIndex();
+    }
+    debugValidate();
+}
+
 // look to see if the curve end intersects an intermediary that intersects the other
 void SkOpSegment::checkEnds() {
     debugValidate();
     SkSTArray<kMissingSpanCount, MissingSpan, true> missingSpans;
     int count = fTs.count();
     for (int index = 0; index < count; ++index) {
         const SkOpSpan& span = fTs[index];
         double otherT = span.fOtherT;
         if (otherT != 0 && otherT != 1) { // only check ends
             continue;
@@ skipping 87 matching lines @@
         }
     }
     if (missingSpans.count() == 0) {
         debugValidate();
         return;
     }
     debugValidate();
     int missingCount = missingSpans.count();
     for (int index = 0; index < missingCount; ++index)  {
         MissingSpan& missing = missingSpans[index];
-        addTPair(missing.fT, missing.fOther, missing.fOtherT, false, missing.fPt);
+        if (this != missing.fOther) {
+            addTPair(missing.fT, missing.fOther, missing.fOtherT, false, missing.fPt);
+        }
     }
     fixOtherTIndex();
     // OPTIMIZATION: this may fix indices more than once. Build an array of unique segments to
     // avoid this
     for (int index = 0; index < missingCount; ++index)  {
         missingSpans[index].fOther->fixOtherTIndex();
     }
     debugValidate();
 }
 
+void SkOpSegment::checkLinks(const SkOpSpan* base,
+        SkTArray<MissingSpan, true>* missingSpans) const {
+    const SkOpSpan* first = fTs.begin();
+    const SkOpSpan* last = fTs.end() - 1;
+    SkASSERT(base >= first && last >= base);
+    const SkOpSegment* other = base->fOther;
+    const SkOpSpan* oFirst = other->fTs.begin();
+    const SkOpSpan* oLast = other->fTs.end() - 1;
+    const SkOpSpan* oSpan = &other->fTs[base->fOtherIndex];
+    const SkOpSpan* test = base;
+    const SkOpSpan* missing = NULL;
+    while (test > first && (--test)->fPt == base->fPt) {
+        CheckOneLink(test, oSpan, oFirst, oLast, &missing, missingSpans);
+    }
+    test = base;
+    while (test < last && (++test)->fPt == base->fPt) {
+        CheckOneLink(test, oSpan, oFirst, oLast, &missing, missingSpans);
+    }
+}
+
+// see if spans with two or more intersections all agree on common t and point values
+void SkOpSegment::checkMultiples() {
+    debugValidate();
+    int index;
+    int end = 0;
+    while (fTs[++end].fT == 0)
+        ;
+    while (fTs[end].fT < 1) {
+        int start = index = end;
+        end = nextExactSpan(index, 1);
+        if (end <= index) {
+            return;  // buffer overflow example triggers this
+        }
+        if (index + 1 == end) {
+            continue;
+        }
+        // force the duplicates to agree on t and pt if not on the end
+        double thisT = fTs[index].fT;
+        const SkPoint& thisPt = fTs[index].fPt;
+        bool aligned = false;
+        while (++index < end) {
+            aligned |= alignSpan(index, thisT, thisPt);
+        }
+        if (aligned) {
+            alignSpanState(start, end);
+        }
+    }
+    debugValidate();
+}
+
+void SkOpSegment::CheckOneLink(const SkOpSpan* test, const SkOpSpan* oSpan,
+        const SkOpSpan* oFirst, const SkOpSpan* oLast, const SkOpSpan** missingPtr,
+        SkTArray<MissingSpan, true>* missingSpans) {
+    SkASSERT(oSpan->fPt == test->fPt);
+    const SkOpSpan* oTest = oSpan;
+    while (oTest > oFirst && (--oTest)->fPt == test->fPt) {
+        if (oTest->fOther == test->fOther && oTest->fOtherT == test->fOtherT) {
+            return;
+        }
+    }
+    oTest = oSpan;
+    while (oTest < oLast && (++oTest)->fPt == test->fPt) {
+        if (oTest->fOther == test->fOther && oTest->fOtherT == test->fOtherT) {
+            return;
+        }
+    }
+    if (*missingPtr) {
+        missingSpans->push_back();
+    }
+    MissingSpan& lastMissing = missingSpans->back();
+    if (*missingPtr) {
+        lastMissing = missingSpans->end()[-2];
+    }
+    *missingPtr = test;
+    lastMissing.fOther = test->fOther;
+    lastMissing.fOtherT = test->fOtherT;
+}
+
 bool SkOpSegment::checkSmall(int index) const {
     if (fTs[index].fSmall) {
         return true;
     }
     double tBase = fTs[index].fT;
     while (index > 0 && precisely_negative(tBase - fTs[--index].fT))
         ;
     return fTs[index].fSmall;
 }
 
+// a pair of curves may turn into coincident lines -- small may be a hint that that happened
+// if a cubic contains a loop, the counts must be adjusted
+void SkOpSegment::checkSmall() {
+    SkSTArray<kMissingSpanCount, MissingSpan, true> missingSpans;
+    const SkOpSpan* beginSpan = fTs.begin();
+    const SkOpSpan* thisSpan = beginSpan - 1;
+    const SkOpSpan* endSpan = fTs.end() - 1;  // last can't be small
+    while (++thisSpan < endSpan) {
+        if (!thisSpan->fSmall) {
+            continue;
+        }
+        if (!thisSpan->fWindValue) {
+            continue;
+        }
+        const SkOpSpan& firstSpan = this->firstSpan(*thisSpan);
+        const SkOpSpan& lastSpan = this->lastSpan(*thisSpan);
+        ptrdiff_t smallCount = &lastSpan - &firstSpan + 1;
+        SkASSERT(1 <= smallCount && smallCount < count());
+        if (smallCount <= 1) {
+            SkASSERT(1 == smallCount);
+            checkSmallCoincidence(firstSpan, NULL);
+            continue;
+        }
+        // at this point, check for missing computed intersections
+        const SkPoint& testPt = firstSpan.fPt;
+        thisSpan = &firstSpan - 1;
+        SkOpSegment* other = NULL;
+        while (++thisSpan <= &lastSpan) {
+            other = thisSpan->fOther;
+            if (other != this) {
+                break;
+            }
+        }
+        SkASSERT(other != this);
+        int oIndex = thisSpan->fOtherIndex;
+        const SkOpSpan& oSpan = other->span(oIndex);
+        const SkOpSpan& oFirstSpan = other->firstSpan(oSpan);
+        const SkOpSpan& oLastSpan = other->lastSpan(oSpan);
+        ptrdiff_t oCount = &oLastSpan - &oFirstSpan + 1;
+        if (fLoop) {
+            int smallCounts[2];
+            SkASSERT(!other->fLoop);  // FIXME: we need more complicated logic for pair of loops
+            if (calcLoopSpanCount(*thisSpan, smallCounts)) {
+                if (smallCounts[0] && oCount != smallCounts[0]) {
+                    SkASSERT(0);  // FIXME: need a working test case to properly code & debug
+                }
+                if (smallCounts[1] && oCount != smallCounts[1]) {
+                    SkASSERT(0);  // FIXME: need a working test case to properly code & debug
+                }
+                goto nextSmallCheck;
+            }
+        }
+        if (other->fLoop) {
+            int otherCounts[2];
+            if (other->calcLoopSpanCount(other->span(oIndex), otherCounts)) {
+                if (otherCounts[0] && otherCounts[0] != smallCount) {
+                    SkASSERT(0);  // FIXME: need a working test case to properly code & debug
+                }
+                if (otherCounts[1] && otherCounts[1] != smallCount) {
+                    SkASSERT(0);  // FIXME: need a working test case to properly code & debug
+                }
+                goto nextSmallCheck;
+            }
+        }
+        if (oCount != smallCount) {  // check if number of pts in this match other
+            MissingSpan& missing = missingSpans.push_back();
+            missing.fOther = NULL;
+            SkDEBUGCODE(sk_bzero(&missing, sizeof(missing)));
+            missing.fPt = testPt;
+            const SkOpSpan& oSpan = other->span(oIndex);
+            if (oCount > smallCount) {
+                missing.fSegment = this;
+                missing.fT = thisSpan->fT;
+                other->checkLinks(&oSpan, &missingSpans);
+            } else {
+                missing.fSegment = other;
+                missing.fT = oSpan.fT;
+                checkLinks(thisSpan, &missingSpans);
+            }
+            if (!missingSpans.back().fOther || missing.fSegment->done()) {
+                missingSpans.pop_back();
+            }
+        }
+nextSmallCheck:
+        thisSpan = &lastSpan;
+    }
+    int missingCount = missingSpans.count();
+    for (int index = 0; index < missingCount; ++index)  {
+        MissingSpan& missing = missingSpans[index];
+        SkOpSegment* missingOther = missing.fOther;
+        // note that add t pair may edit span arrays, so prior pointers to spans are no longer valid
+        if (!missing.fSegment->addTPair(missing.fT, missingOther, missing.fOtherT, false,
+                missing.fPt)) {
+            continue;
+        }
+        int otherTIndex = missingOther->findT(missing.fOtherT, missing.fSegment);
+        const SkOpSpan& otherSpan = missingOther->span(otherTIndex);
+        if (otherSpan.fSmall) {
+            const SkOpSpan* nextSpan = &otherSpan;
+            do {
+                ++nextSpan;
+            } while (nextSpan->fSmall);
+            missing.fSegment->addTCoincident(missing.fPt, nextSpan->fPt, nextSpan->fT,
+                    missingOther);
+        } else if (otherSpan.fT > 0) {
+            const SkOpSpan* priorSpan = &otherSpan;
+            do {
+                --priorSpan;
+            } while (priorSpan->fT == otherSpan.fT);
+            if (priorSpan->fSmall) {
+                missing.fSegment->addTCancel(missing.fPt, priorSpan->fPt, missingOther);
+            }
+        }
+    }
+    // OPTIMIZATION: this may fix indices more than once. Build an array of unique segments to
+    // avoid this
+    for (int index = 0; index < missingCount; ++index)  {
+        MissingSpan& missing = missingSpans[index];
+        missing.fSegment->fixOtherTIndex();
+        missing.fOther->fixOtherTIndex();
+    }
+    debugValidate();
+}
+
+void SkOpSegment::checkSmallCoincidence(const SkOpSpan& span,
+        SkTArray<MissingSpan, true>* checkMultiple) {
+    SkASSERT(span.fSmall);
+    SkASSERT(span.fWindValue);
+    SkASSERT(&span < fTs.end() - 1);
+    const SkOpSpan* next = &span + 1;
+    SkASSERT(!next->fSmall || checkMultiple);
+    if (checkMultiple) {
+        while (next->fSmall) {
+            ++next;
+            SkASSERT(next < fTs.end());
+        }
+    }
+    SkOpSegment* other = span.fOther;
+    while (other != next->fOther) {
+        if (!checkMultiple) {
+            return;
+        }
+        const SkOpSpan* test = next + 1;
+        if (test == fTs.end()) {
+            return;
+        }
+        if (test->fPt != next->fPt || !precisely_equal(test->fT, next->fT)) {
+            return;
+        }
+        next = test;
+    }
+    SkASSERT(span.fT < next->fT);
+    int oStartIndex = other->findExactT(span.fOtherT, this);
+    int oEndIndex = other->findExactT(next->fOtherT, this);
+    // FIXME: be overly conservative by limiting this to the caller that allows multiple smalls
+    if (!checkMultiple || fVerb != SkPath::kLine_Verb || other->fVerb != SkPath::kLine_Verb) {
+        SkPoint mid = ptAtT((span.fT + next->fT) / 2);
+        const SkOpSpan& oSpanStart = other->fTs[oStartIndex];
+        const SkOpSpan& oSpanEnd = other->fTs[oEndIndex];
+        SkPoint oMid = other->ptAtT((oSpanStart.fT + oSpanEnd.fT) / 2);
+        if (!SkDPoint::ApproximatelyEqual(mid, oMid)) {
+            return;
+        }
+    }
+    // FIXME: again, be overly conservative to avoid breaking existing tests
+    const SkOpSpan& oSpan = oStartIndex < oEndIndex ? other->fTs[oStartIndex]
+            : other->fTs[oEndIndex];
+    if (checkMultiple && !oSpan.fSmall) {
+        return;
+    }
+    SkASSERT(oSpan.fSmall);
+    if (oStartIndex < oEndIndex) {
+        addTCoincident(span.fPt, next->fPt, next->fT, other);
+    } else {
+        addTCancel(span.fPt, next->fPt, other);
+    }
+    if (!checkMultiple) {
+        return;
+    }
+    // check to see if either segment is coincident with a third segment -- if it is, and if
+    // the opposite segment is not already coincident with the third, make it so
+    // OPTIMIZE: to make this check easier, add coincident and cancel could set a coincident bit
+    if (span.fWindValue != 1 || span.fOppValue != 0) {
+//        start here;
+        // iterate through the spans, looking for the third coincident case
+        // if we find one, we need to return state to the caller so that the indices can be fixed
+        // this also suggests that all of this function is fragile since it relies on a valid index
+    }
+    // probably should make this a common function rather than copy/paste code
+    if (oSpan.fWindValue != 1 || oSpan.fOppValue != 0) {
+        const SkOpSpan* oTest = &oSpan;
+        while (--oTest >= other->fTs.begin()) {
+            if (oTest->fPt != oSpan.fPt || !precisely_equal(oTest->fT, oSpan.fT)) {
+                break;
+            }
+            SkOpSegment* testOther = oTest->fOther;
+            SkASSERT(testOther != this);
+            // look in both directions to see if there is a coincident span
+            const SkOpSpan* tTest = testOther->fTs.begin();
+            for (int testIndex = 0; testIndex < testOther->count(); ++testIndex) {
+                if (tTest->fPt != span.fPt) {
+                    ++tTest;
+                    continue;
+                }
+                if (testOther->verb() != SkPath::kLine_Verb
+                        || other->verb() != SkPath::kLine_Verb) {
+                    SkPoint mid = ptAtT((span.fT + next->fT) / 2);
+                    SkPoint oMid = other->ptAtT((oTest->fOtherT + tTest->fT) / 2);
+                    if (!SkDPoint::ApproximatelyEqual(mid, oMid)) {
+                        continue;
+                    }
+                } 
+#if DEBUG_CONCIDENT
+                SkDebugf("%s coincident found=%d %1.9g %1.9g\n", __FUNCTION__, testOther->fID,
+                        oTest->fOtherT, tTest->fT);
+#endif
+                if (tTest->fT < oTest->fOtherT) {
+                    addTCoincident(span.fPt, next->fPt, next->fT, testOther);
+                } else {
+                    addTCancel(span.fPt, next->fPt, testOther);
+                }
+                MissingSpan missing;
+                missing.fSegment = testOther;
+                checkMultiple->push_back(missing);
+                break;
+            }
+        }
+        oTest = &oSpan;
+        while (++oTest < other->fTs.end()) {
+            if (oTest->fPt != oSpan.fPt || !precisely_equal(oTest->fT, oSpan.fT)) {
+                break;
+            }
+
+        }
+    }
+}
+
 // if pair of spans on either side of tiny have the same end point and mid point, mark
 // them as parallel
-// OPTIMIZATION : mark the segment to note that some span is tiny
 void SkOpSegment::checkTiny() {
     SkSTArray<kMissingSpanCount, MissingSpan, true> missingSpans;
     SkOpSpan* thisSpan = fTs.begin() - 1;
     const SkOpSpan* endSpan = fTs.end() - 1;  // last can't be tiny
     while (++thisSpan < endSpan) {
         if (!thisSpan->fTiny) {
             continue;
         }
         SkOpSpan* nextSpan = thisSpan + 1;
         double thisT = thisSpan->fT;
@@ skipping 44 matching lines @@
                 missing.fPt = thisSpan->fPt;
             }
         }
     }
     int missingCount = missingSpans.count();
     if (!missingCount) {
         return;
     }
     for (int index = 0; index < missingCount; ++index)  {
         MissingSpan& missing = missingSpans[index];
-        missing.fSegment->addTPair(missing.fT, missing.fOther, missing.fOtherT, false, missing.fPt);
+        if (missing.fSegment != missing.fOther) {
+            missing.fSegment->addTPair(missing.fT, missing.fOther, missing.fOtherT, false,
+                    missing.fPt);
+        }
     }
+    // OPTIMIZE: consolidate to avoid multiple calls to fix index
     for (int index = 0; index < missingCount; ++index)  {
         MissingSpan& missing = missingSpans[index];
         missing.fSegment->fixOtherTIndex();
         missing.fOther->fixOtherTIndex();
     }
 }
 
 bool SkOpSegment::findCoincidentMatch(const SkOpSpan* span, const SkOpSegment* other, int oStart,
         int oEnd, int step, SkPoint* startPt, SkPoint* endPt, double* endT) const {
     SkASSERT(span->fT == 0 || span->fT == 1);
@@ skipping 51 matching lines @@
             return false;
         }
         if (otherSpan == lastSpan) {
             break;
         }
         otherSpan += step;
     } while (otherSpan->fT == refT || otherSpan->fPt == refPt);
     return false;
 }
 
+int SkOpSegment::findEndSpan(int endIndex) const {
+    const SkOpSpan* span = &fTs[--endIndex];
+    const SkPoint& lastPt = span->fPt;
+    double endT = span->fT;
+    do {
+        span = &fTs[--endIndex];
+    } while (SkDPoint::ApproximatelyEqual(span->fPt, lastPt) && (span->fT == endT || span->fTiny));
+    return endIndex + 1;
+}
+
 /*
  The M and S variable name parts stand for the operators.
    Mi stands for Minuend (see wiki subtraction, analogous to difference)
    Su stands for Subtrahend
  The Opp variable name part designates that the value is for the Opposite operator.
  Opposite values result from combining coincident spans.
  */
 SkOpSegment* SkOpSegment::findNextOp(SkTDArray<SkOpSpan*>* chase, int* nextStart, int* nextEnd,
                                      bool* unsortable, SkPathOp op, const int xorMiMask,
                                      const int xorSuMask) {
@@ skipping 25 matching lines @@
         do {
             *nextEnd += step;
         } while (precisely_zero(startT - other->fTs[*nextEnd].fT));
         SkASSERT(step < 0 ? *nextEnd >= 0 : *nextEnd < other->fTs.count());
         if (other->isTiny(SkMin32(*nextStart, *nextEnd))) {
             *unsortable = true;
             return NULL;
         }
         return other;
     }
-    // more than one viable candidate -- measure angles to find best
-    SkSTArray<SkOpAngle::kStackBasedCount, SkOpAngle, true> angles;
     SkASSERT(startIndex - endIndex != 0);
     SkASSERT((startIndex - endIndex < 0) ^ (step < 0));
-    SkSTArray<SkOpAngle::kStackBasedCount, SkOpAngle*, true> sorted;
-    int calcWinding = computeSum(startIndex, end, SkOpAngle::kBinaryOpp, &angles, &sorted);
+    // more than one viable candidate -- measure angles to find best
+
+    int calcWinding = computeSum(startIndex, end, SkOpAngle::kBinaryOpp);
     bool sortable = calcWinding != SK_NaN32;
-    if (sortable && sorted.count() == 0) {
-        // if no edge has a computed winding sum, we can go no further
-        *unsortable = true;
-        return NULL;
-    }
-    int angleCount = angles.count();
-    int firstIndex = findStartingEdge(sorted, startIndex, end);
-    SkASSERT(!sortable || firstIndex >= 0);
-#if DEBUG_SORT
-    debugShowSort(__FUNCTION__, sorted, firstIndex, sortable);
-#endif
     if (!sortable) {
         *unsortable = true;
         return NULL;
     }
-    SkASSERT(sorted[firstIndex]->segment() == this);
-#if DEBUG_WINDING
-    SkDebugf("%s firstIndex=[%d] sign=%d\n", __FUNCTION__, firstIndex,
-            sorted[firstIndex]->sign());
+    SkOpAngle* angle = spanToAngle(end, startIndex);
+    if (angle->unorderable()) {
+        *unsortable = true;
+        return NULL;
+    }
+#if DEBUG_SORT
+    SkDebugf("%s\n", __FUNCTION__);
+    angle->debugLoop();
 #endif
     int sumMiWinding = updateWinding(endIndex, startIndex);
     int sumSuWinding = updateOppWinding(endIndex, startIndex);
     if (operand()) {
         SkTSwap<int>(sumMiWinding, sumSuWinding);
     }
-    int nextIndex = firstIndex + 1;
-    int lastIndex = firstIndex != 0 ? firstIndex : angleCount;
+    SkOpAngle* nextAngle = angle->next();
     const SkOpAngle* foundAngle = NULL;
     bool foundDone = false;
     // iterate through the angle, and compute everyone's winding
     SkOpSegment* nextSegment;
     int activeCount = 0;
     do {
-        SkASSERT(nextIndex != firstIndex);
-        if (nextIndex == angleCount) {
-            nextIndex = 0;
-        }
-        const SkOpAngle* nextAngle = sorted[nextIndex];
         nextSegment = nextAngle->segment();
-        int maxWinding, sumWinding, oppMaxWinding, oppSumWinding;
         bool activeAngle = nextSegment->activeOp(xorMiMask, xorSuMask, nextAngle->start(),
-                nextAngle->end(), op, &sumMiWinding, &sumSuWinding,
-                &maxWinding, &sumWinding, &oppMaxWinding, &oppSumWinding);
+                nextAngle->end(), op, &sumMiWinding, &sumSuWinding);
         if (activeAngle) {
             ++activeCount;
             if (!foundAngle || (foundDone && activeCount & 1)) {
                 if (nextSegment->isTiny(nextAngle)) {
                     *unsortable = true;
                     return NULL;
                 }
                 foundAngle = nextAngle;
                 foundDone = nextSegment->done(nextAngle);
             }
         }
         if (nextSegment->done()) {
             continue;
         }
         if (nextSegment->isTiny(nextAngle)) {
             continue;
         }
         if (!activeAngle) {
             nextSegment->markAndChaseDoneBinary(nextAngle->start(), nextAngle->end());
         }
         SkOpSpan* last = nextAngle->lastMarked();
         if (last) {
+            SkASSERT(!SkPathOpsDebug::ChaseContains(*chase, last));
             *chase->append() = last;
 #if DEBUG_WINDING
             SkDebugf("%s chase.append id=%d windSum=%d small=%d\n", __FUNCTION__,
                     last->fOther->fTs[last->fOtherIndex].fOther->debugID(), last->fWindSum,
                     last->fSmall);
 #endif
         }
-    } while (++nextIndex != lastIndex);
+    } while ((nextAngle = nextAngle->next()) != angle);
+#if DEBUG_ANGLE
+    if (foundAngle) {
+        foundAngle->debugSameAs(foundAngle);
+    }
+#endif
+
     markDoneBinary(SkMin32(startIndex, endIndex));
     if (!foundAngle) {
         return NULL;
     }
     *nextStart = foundAngle->start();
     *nextEnd = foundAngle->end();
     nextSegment = foundAngle->segment();
 #if DEBUG_WINDING
     SkDebugf("%s from:[%d] to:[%d] start=%d end=%d\n",
             __FUNCTION__, debugID(), nextSegment->debugID(), *nextStart, *nextEnd);
@@ skipping 31 matching lines @@
         do {
             *nextEnd += step;
         } while (precisely_zero(startT - other->fTs[*nextEnd].fT));
         SkASSERT(step < 0 ? *nextEnd >= 0 : *nextEnd < other->fTs.count());
         if (other->isTiny(SkMin32(*nextStart, *nextEnd))) {
             *unsortable = true;
             return NULL;
         }
         return other;
     }
-    // more than one viable candidate -- measure angles to find best
-    SkSTArray<SkOpAngle::kStackBasedCount, SkOpAngle, true> angles;
     SkASSERT(startIndex - endIndex != 0);
     SkASSERT((startIndex - endIndex < 0) ^ (step < 0));
-    SkSTArray<SkOpAngle::kStackBasedCount, SkOpAngle*, true> sorted;
-    int calcWinding = computeSum(startIndex, end, SkOpAngle::kUnaryWinding, &angles, &sorted);
+    // more than one viable candidate -- measure angles to find best
+
+    int calcWinding = computeSum(startIndex, end, SkOpAngle::kUnaryWinding);
     bool sortable = calcWinding != SK_NaN32;
-    int angleCount = angles.count();
-    int firstIndex = findStartingEdge(sorted, startIndex, end);
-    SkASSERT(!sortable || firstIndex >= 0);
-#if DEBUG_SORT
-    debugShowSort(__FUNCTION__, sorted, firstIndex, sortable);
-#endif
     if (!sortable) {
         *unsortable = true;
         return NULL;
     }
-    SkASSERT(sorted[firstIndex]->segment() == this);
-#if DEBUG_WINDING
-    SkDebugf("%s firstIndex=[%d] sign=%d\n", __FUNCTION__, firstIndex,
-            sorted[firstIndex]->sign());
+    SkOpAngle* angle = spanToAngle(end, startIndex);
+#if DEBUG_SORT
+    SkDebugf("%s\n", __FUNCTION__);
+    angle->debugLoop();
 #endif
     int sumWinding = updateWinding(endIndex, startIndex);
-    int nextIndex = firstIndex + 1;
-    int lastIndex = firstIndex != 0 ? firstIndex : angleCount;
+    SkOpAngle* nextAngle = angle->next();
     const SkOpAngle* foundAngle = NULL;
     bool foundDone = false;
-    // iterate through the angle, and compute everyone's winding
     SkOpSegment* nextSegment;
     int activeCount = 0;
     do {
-        SkASSERT(nextIndex != firstIndex);
-        if (nextIndex == angleCount) {
-            nextIndex = 0;
-        }
-        const SkOpAngle* nextAngle = sorted[nextIndex];
         nextSegment = nextAngle->segment();
-        int maxWinding;
         bool activeAngle = nextSegment->activeWinding(nextAngle->start(), nextAngle->end(),
-                &maxWinding, &sumWinding);
+                &sumWinding);
         if (activeAngle) {
             ++activeCount;
             if (!foundAngle || (foundDone && activeCount & 1)) {
                 if (nextSegment->isTiny(nextAngle)) {
                     *unsortable = true;
                     return NULL;
                 }
                 foundAngle = nextAngle;
                 foundDone = nextSegment->done(nextAngle);
             }
         }
         if (nextSegment->done()) {
             continue;
         }
         if (nextSegment->isTiny(nextAngle)) {
             continue;
         }
         if (!activeAngle) {
             nextSegment->markAndChaseDoneUnary(nextAngle->start(), nextAngle->end());
         }
         SkOpSpan* last = nextAngle->lastMarked();
         if (last) {
+            SkASSERT(!SkPathOpsDebug::ChaseContains(*chase, last));
+            // assert here that span isn't already in array
             *chase->append() = last;
 #if DEBUG_WINDING
             SkDebugf("%s chase.append id=%d windSum=%d small=%d\n", __FUNCTION__,
                     last->fOther->fTs[last->fOtherIndex].fOther->debugID(), last->fWindSum,
                     last->fSmall);
 #endif
         }
-    } while (++nextIndex != lastIndex);
+    } while ((nextAngle = nextAngle->next()) != angle);
     markDoneUnary(SkMin32(startIndex, endIndex));
     if (!foundAngle) {
         return NULL;
     }
     *nextStart = foundAngle->start();
     *nextEnd = foundAngle->end();
     nextSegment = foundAngle->segment();
 #if DEBUG_WINDING
     SkDebugf("%s from:[%d] to:[%d] start=%d end=%d\n",
             __FUNCTION__, debugID(), nextSegment->debugID(), *nextStart, *nextEnd);
  #endif
     return nextSegment;
 }
 
 SkOpSegment* SkOpSegment::findNextXor(int* nextStart, int* nextEnd, bool* unsortable) {
     const int startIndex = *nextStart;
     const int endIndex = *nextEnd;
     SkASSERT(startIndex != endIndex);
     SkDEBUGCODE(int count = fTs.count());
     SkASSERT(startIndex < endIndex ? startIndex < count - 1 : startIndex > 0);
     int step = SkSign32(endIndex - startIndex);
     int end = nextExactSpan(startIndex, step);
     SkASSERT(end >= 0);
     SkOpSpan* endSpan = &fTs[end];
     SkOpSegment* other;
-    if (isSimple(end)) {
+// Detect cases where all the ends canceled out (e.g.,
+// there is no angle) and therefore there's only one valid connection
+    if (isSimple(end) || !spanToAngle(end, startIndex)) {
 #if DEBUG_WINDING
         SkDebugf("%s simple\n", __FUNCTION__);
 #endif
         int min = SkMin32(startIndex, endIndex);
         if (fTs[min].fDone) {
             return NULL;
         }
         markDone(min, 1);
         other = endSpan->fOther;
         *nextStart = endSpan->fOtherIndex;
@@ skipping 13 matching lines @@
             if (other->fTs[SkMin32(*nextStart, *nextEnd)].fWindValue) {
                 break;
             }
             SkASSERT(firstLoop);
             SkDEBUGCODE(firstLoop = false;)
             step = -step;
         } while (true);
         SkASSERT(step < 0 ? *nextEnd >= 0 : *nextEnd < other->fTs.count());
         return other;
     }
-    SkSTArray<SkOpAngle::kStackBasedCount, SkOpAngle, true> angles;
     SkASSERT(startIndex - endIndex != 0);
     SkASSERT((startIndex - endIndex < 0) ^ (step < 0));
-    SkSTArray<SkOpAngle::kStackBasedCount, SkOpAngle*, true> sorted;
-    int calcWinding = computeSum(startIndex, end, SkOpAngle::kUnaryXor, &angles, &sorted);
-    bool sortable = calcWinding != SK_NaN32;
-    int angleCount = angles.count();
-    int firstIndex = findStartingEdge(sorted, startIndex, end);
-    SkASSERT(!sortable || firstIndex >= 0);
+    // parallel block above with presorted version
+    SkOpAngle* angle = spanToAngle(end, startIndex);
+    SkASSERT(angle);
 #if DEBUG_SORT
-    debugShowSort(__FUNCTION__, sorted, firstIndex, 0, 0, sortable);
+    SkDebugf("%s\n", __FUNCTION__);
+    angle->debugLoop();
 #endif
-    if (!sortable) {
-        *unsortable = true;
-        return NULL;
-    }
-    SkASSERT(sorted[firstIndex]->segment() == this);
-#if DEBUG_WINDING
-    SkDebugf("%s firstIndex=[%d] sign=%d\n", __FUNCTION__, firstIndex,
-            sorted[firstIndex]->sign());
-#endif
-    int nextIndex = firstIndex + 1;
-    int lastIndex = firstIndex != 0 ? firstIndex : angleCount;
+    SkOpAngle* nextAngle = angle->next();
     const SkOpAngle* foundAngle = NULL;
     bool foundDone = false;
     SkOpSegment* nextSegment;
     int activeCount = 0;
     do {
-        SkASSERT(nextIndex != firstIndex);
-        if (nextIndex == angleCount) {
-            nextIndex = 0;
-        }
-        const SkOpAngle* nextAngle = sorted[nextIndex];
         nextSegment = nextAngle->segment();
         ++activeCount;
         if (!foundAngle || (foundDone && activeCount & 1)) {
             if (nextSegment->isTiny(nextAngle)) {
                 *unsortable = true;
                 return NULL;
             }
             foundAngle = nextAngle;
-            foundDone = nextSegment->done(nextAngle);
+            if (!(foundDone = nextSegment->done(nextAngle))) {
+                break;
+            }
         }
-        if (nextSegment->done()) {
-            continue;
-        }
-    } while (++nextIndex != lastIndex);
+        nextAngle = nextAngle->next();
+    } while (nextAngle != angle);
     markDone(SkMin32(startIndex, endIndex), 1);
     if (!foundAngle) {
         return NULL;
     }
     *nextStart = foundAngle->start();
     *nextEnd = foundAngle->end();
     nextSegment = foundAngle->segment();
 #if DEBUG_WINDING
     SkDebugf("%s from:[%d] to:[%d] start=%d end=%d\n",
             __FUNCTION__, debugID(), nextSegment->debugID(), *nextStart, *nextEnd);
  #endif
     return nextSegment;
 }
 
-int SkOpSegment::findStartingEdge(const SkTArray<SkOpAngle*, true>& sorted, int start, int end) {
-    int angleCount = sorted.count();
-    int firstIndex = -1;
-    for (int angleIndex = 0; angleIndex < angleCount; ++angleIndex) {
-        const SkOpAngle* angle = sorted[angleIndex];
-        if (angle->segment() == this && angle->start() == end &&
-                angle->end() == start) {
-            firstIndex = angleIndex;
-            break;
+int SkOpSegment::findStartSpan(int startIndex) const {
+    int index = startIndex;
+    const SkOpSpan* span = &fTs[index];
+    const SkPoint& firstPt = span->fPt;
+    double firstT = span->fT;
+    do {
+        if (index > 0) {
+            if (span->fT != firstT) {
+                break;
+            }
+            if (!SkDPoint::ApproximatelyEqual(firstPt, fTs[index].fPt)) {
+                return -1;
+            }
         }
-    }
-    return firstIndex;
+        ++index;
+        if (span->fTiny) {
+            if (!SkDPoint::ApproximatelyEqual(firstPt, fTs[index].fPt)) {
+                return -1;
+            }
+            firstT = fTs[index++].fT;
+        }
+        span = &fTs[index];
+    } while (true);
+    return index;
 }
 
-int SkOpSegment::findT(double t, const SkOpSegment* match) const {
+int SkOpSegment::findExactT(double t, const SkOpSegment* match) const {
     int count = this->count();
     for (int index = 0; index < count; ++index) {
         const SkOpSpan& span = fTs[index];
         if (span.fT == t && span.fOther == match) {
             return index;
         }
     }
     SkASSERT(0);
     return -1;
 }
 
-// FIXME: either:
-// a) mark spans with either end unsortable as done, or
-// b) rewrite findTop / findTopSegment / findTopContour to iterate further
-//    when encountering an unsortable span
+int SkOpSegment::findT(double t, const SkOpSegment* match) const {
+    int count = this->count();
+    for (int index = 0; index < count; ++index) {
+        const SkOpSpan& span = fTs[index];
+        if (approximately_equal_orderable(span.fT, t) && span.fOther == match) {
+            return index;
+        }
+    }
+    SkASSERT(0);
+    return -1;
+}
 
-// OPTIMIZATION : for a pair of lines, can we compute points at T (cached)
-// and use more concise logic like the old edge walker code?
-// FIXME: this needs to deal with coincident edges
-SkOpSegment* SkOpSegment::findTop(int* tIndexPtr, int* endIndexPtr, bool* unsortable,
-                                  bool onlySortable) {
+SkOpSegment* SkOpSegment::findTop(int* tIndexPtr, int* endIndexPtr, bool* unsortable) {
     // iterate through T intersections and return topmost
     // topmost tangent from y-min to first pt is closer to horizontal
     SkASSERT(!done());
     int firstT = -1;
-    /* SkPoint topPt = */ activeLeftTop(onlySortable, &firstT);
+    /* SkPoint topPt = */ activeLeftTop(true, &firstT);
     if (firstT < 0) {
         *unsortable = true;
         firstT = 0;
         while (fTs[firstT].fDone) {
             SkASSERT(firstT < fTs.count());
             ++firstT;
         }
         *tIndexPtr = firstT;
         *endIndexPtr = nextExactSpan(firstT, 1);
         return this;
     }
     // sort the edges to find the leftmost
     int step = 1;
-    int end = nextSpan(firstT, step);
-    if (end == -1) {
+    int end;
+    if (span(firstT).fDone || (end = nextSpan(firstT, step)) == -1) {
         step = -1;
         end = nextSpan(firstT, step);
         SkASSERT(end != -1);
     }
     // if the topmost T is not on end, or is three-way or more, find left
     // look for left-ness from tLeft to firstT (matching y of other)
-    SkSTArray<SkOpAngle::kStackBasedCount, SkOpAngle, true> angles;
     SkASSERT(firstT - end != 0);
-    addTwoAngles(end, firstT, &angles);
-    if (!buildAngles(firstT, &angles, true) && onlySortable) {
-//        *unsortable = true;
-//        return NULL;
+    SkOpAngle* markAngle = spanToAngle(firstT, end);
+    if (!markAngle) {
+        markAngle = addSingletonAngles(firstT, step);
     }
-    SkSTArray<SkOpAngle::kStackBasedCount, SkOpAngle*, true> sorted;
-    bool sortable = SortAngles(angles, &sorted, SkOpSegment::kMayBeUnordered_SortAngleKind);
-    if (onlySortable && !sortable) {
-        *unsortable = true;
-        return NULL;
+    markAngle->markStops();
+    const SkOpAngle* baseAngle = markAngle->findFirst();
+    if (!baseAngle) {
+        return NULL;  // nothing to do
     }
-    int first = SK_MaxS32;
     SkScalar top = SK_ScalarMax;
-    int count = sorted.count();
-    for (int index = 0; index < count; ++index) {
-        const SkOpAngle* angle = sorted[index];
-        if (onlySortable && angle->unorderable()) {
-            continue;
+    const SkOpAngle* firstAngle = NULL;
+    const SkOpAngle* angle = baseAngle;
+    do {
+        if (!angle->unorderable()) {
+            SkOpSegment* next = angle->segment();
+            SkPathOpsBounds bounds;
+            next->subDivideBounds(angle->end(), angle->start(), &bounds);
+            if (approximately_greater(top, bounds.fTop)) {
+                top = bounds.fTop;
+                firstAngle = angle;
+            }
         }
-        SkOpSegment* next = angle->segment();
-        SkPathOpsBounds bounds;
-        next->subDivideBounds(angle->end(), angle->start(), &bounds);
-        if (approximately_greater(top, bounds.fTop)) {
-            top = bounds.fTop;
-            first = index;
-        }
-    }
-    SkASSERT(first < SK_MaxS32);
-#if DEBUG_SORT  // || DEBUG_SWAP_TOP
-    sorted[first]->segment()->debugShowSort(__FUNCTION__, sorted, first, 0, 0, sortable);
+        angle = angle->next();
+    } while (angle != baseAngle);
+    SkASSERT(firstAngle);
+#if DEBUG_SORT
+    SkDebugf("%s\n", __FUNCTION__);
+    firstAngle->debugLoop();
 #endif
     // skip edges that have already been processed
-    firstT = first - 1;
+    angle = firstAngle;
     SkOpSegment* leftSegment;
     do {
-        if (++firstT == count) {
-            firstT = 0;
-        }
-        const SkOpAngle* angle = sorted[firstT];
-        SkASSERT(!onlySortable || !angle->unsortable());
+//        SkASSERT(!angle->unsortable());
         leftSegment = angle->segment();
         *tIndexPtr = angle->end();
         *endIndexPtr = angle->start();
+        angle = angle->next();
     } while (leftSegment->fTs[SkMin32(*tIndexPtr, *endIndexPtr)].fDone);
     if (leftSegment->verb() >= SkPath::kQuad_Verb) {
         const int tIndex = *tIndexPtr;
         const int endIndex = *endIndexPtr;
         if (!leftSegment->clockwise(tIndex, endIndex)) {
             bool swap = !leftSegment->monotonicInY(tIndex, endIndex)
                     && !leftSegment->serpentine(tIndex, endIndex);
     #if DEBUG_SWAP_TOP
             SkDebugf("%s swap=%d serpentine=%d containedByEnds=%d monotonic=%d\n", __FUNCTION__,
                     swap,
                     leftSegment->serpentine(tIndex, endIndex),
                     leftSegment->controlsContainedByEnds(tIndex, endIndex),
                     leftSegment->monotonicInY(tIndex, endIndex));
     #endif
             if (swap) {
     // FIXME: I doubt it makes sense to (necessarily) swap if the edge was not the first
     // sorted but merely the first not already processed (i.e., not done)
                 SkTSwap(*tIndexPtr, *endIndexPtr);
             }
         }
     }
     SkASSERT(!leftSegment->fTs[SkMin32(*tIndexPtr, *endIndexPtr)].fTiny);
     return leftSegment;
 }
 
+int SkOpSegment::firstActive(int tIndex) const {
+    while (fTs[tIndex].fTiny) {
+        SkASSERT(!isCanceled(tIndex));
+        ++tIndex;
+    }
+    return tIndex;
+}
+
 // FIXME: not crazy about this
 // when the intersections are performed, the other index is into an
 // incomplete array. As the array grows, the indices become incorrect
 // while the following fixes the indices up again, it isn't smart about
 // skipping segments whose indices are already correct
 // assuming we leave the code that wrote the index in the first place
 // FIXME: if called after remove, this needs to correct tiny
 void SkOpSegment::fixOtherTIndex() {
     int iCount = fTs.count();
     for (int i = 0; i < iCount; ++i) {
@@ skipping 13 matching lines @@
         SkASSERT(iSpan.fOtherIndex >= 0);
     }
 }
 
 void SkOpSegment::init(const SkPoint pts[], SkPath::Verb verb, bool operand, bool evenOdd) {
     fDoneSpans = 0;
     fOperand = operand;
     fXor = evenOdd;
     fPts = pts;
     fVerb = verb;
+    fLoop = fSmall = fTiny = false;
 }
 
-void SkOpSegment::initWinding(int start, int end) {
+void SkOpSegment::initWinding(int start, int end, SkOpAngle::IncludeType angleIncludeType) {
     int local = spanSign(start, end);
-    int oppLocal = oppSign(start, end);
-    (void) markAndChaseWinding(start, end, local, oppLocal);
+    if (angleIncludeType == SkOpAngle::kBinarySingle) {
+        int oppLocal = oppSign(start, end);
+        (void) markAndChaseWinding(start, end, local, oppLocal);
     // OPTIMIZATION: the reverse mark and chase could skip the first marking
-    (void) markAndChaseWinding(end, start, local, oppLocal);
+        (void) markAndChaseWinding(end, start, local, oppLocal);
+    } else {
+        (void) markAndChaseWinding(start, end, local);
+    // OPTIMIZATION: the reverse mark and chase could skip the first marking
+        (void) markAndChaseWinding(end, start, local);
+    }
 }
 
 /*
 when we start with a vertical intersect, we try to use the dx to determine if the edge is to
 the left or the right of vertical. This determines if we need to add the span's
 sign or not. However, this isn't enough.
 If the supplied sign (winding) is zero, then we didn't hit another vertical span, so dx is needed.
 If there was a winding, then it may or may not need adjusting. If the span the winding was borrowed
 from has the same x direction as this span, the winding should change. If the dx is opposite, then
 the same winding is shared by both.
 */
 void SkOpSegment::initWinding(int start, int end, double tHit, int winding, SkScalar hitDx,
                               int oppWind, SkScalar hitOppDx) {
     SkASSERT(hitDx || !winding);
     SkScalar dx = (*CurveSlopeAtT[SkPathOpsVerbToPoints(fVerb)])(fPts, tHit).fX;
     SkASSERT(dx);
     int windVal = windValue(SkMin32(start, end));
 #if DEBUG_WINDING_AT_T
     SkDebugf("%s oldWinding=%d hitDx=%c dx=%c windVal=%d", __FUNCTION__, winding,
             hitDx ? hitDx > 0 ? '+' : '-' : '0', dx > 0 ? '+' : '-', windVal);
 #endif
-    if (!winding) {
-        winding = dx < 0 ? windVal : -windVal;
-    } else if (winding * dx < 0) {
-        int sideWind = winding + (dx < 0 ? windVal : -windVal);
-        if (abs(winding) < abs(sideWind)) {
-            winding = sideWind;
-        }
+    int sideWind = winding + (dx < 0 ? windVal : -windVal);
+    if (abs(winding) < abs(sideWind)) {
+        winding = sideWind;
     }
 #if DEBUG_WINDING_AT_T
     SkDebugf(" winding=%d\n", winding);
 #endif
     SkDEBUGCODE(int oppLocal = oppSign(start, end));
     SkASSERT(hitOppDx || !oppWind || !oppLocal);
     int oppWindVal = oppValue(SkMin32(start, end));
     if (!oppWind) {
         oppWind = dx < 0 ? oppWindVal : -oppWindVal;
     } else if (hitOppDx * dx >= 0) {
         int oppSideWind = oppWind + (dx < 0 ? oppWindVal : -oppWindVal);
         if (abs(oppWind) < abs(oppSideWind)) {
             oppWind = oppSideWind;
         }
     }
     (void) markAndChaseWinding(start, end, winding, oppWind);
     // OPTIMIZATION: the reverse mark and chase could skip the first marking
     (void) markAndChaseWinding(end, start, winding, oppWind);
 }
 
+bool SkOpSegment::inLoop(const SkOpAngle* baseAngle, int spanCount, int* indexPtr) const {
+    if (!baseAngle->inLoop()) {
+        return false;
+    }
+    int index = *indexPtr;
+    int froIndex = fTs[index].fFromAngleIndex;
+    int toIndex = fTs[index].fToAngleIndex;
+    while (++index < spanCount) {
+        int nextFro = fTs[index].fFromAngleIndex;
+        int nextTo = fTs[index].fToAngleIndex;
+        if (froIndex != nextFro || toIndex != nextTo) {
+            break;
+        }
+    }
+    *indexPtr = index;
+    return true;
+}
+
 // OPTIMIZE: successive calls could start were the last leaves off
 // or calls could specialize to walk forwards or backwards
 bool SkOpSegment::isMissing(double startT, const SkPoint& pt) const {
-    size_t tCount = fTs.count();
-    for (size_t index = 0; index < tCount; ++index) {
+    int tCount = fTs.count();
+    for (int index = 0; index < tCount; ++index) {
         const SkOpSpan& span = fTs[index];
         if (approximately_zero(startT - span.fT) && pt == span.fPt) {
             return false;
         }
     }
     return true;
 }
 
 bool SkOpSegment::isSimple(int end) const {
+#if 1
     int count = fTs.count();
     if (count == 2) {
         return true;
     }
     double t = fTs[end].fT;
     if (approximately_less_than_zero(t)) {
         return !approximately_less_than_zero(fTs[1].fT);
     }
     if (approximately_greater_than_one(t)) {
         return !approximately_greater_than_one(fTs[count - 2].fT);
     }
     return false;
+#else
+    // OPTIMIZE: code could be rejiggered to use this simpler test. To make this work, a little
+    // more logic is required to ignore the dangling canceled out spans
+    const SkOpSpan& span = fTs[end];
+    return span.fFromAngleIndex < 0 && span.fToAngleIndex < 0;
+#endif
+}
+
+bool SkOpSegment::isSmall(const SkOpAngle* angle) const {
+    int start = angle->start();
+    int end = angle->end();
+    const SkOpSpan& mSpan = fTs[SkMin32(start, end)];
+    return mSpan.fSmall;
 }
 
 bool SkOpSegment::isTiny(const SkOpAngle* angle) const {
     int start = angle->start();
     int end = angle->end();
     const SkOpSpan& mSpan = fTs[SkMin32(start, end)];
     return mSpan.fTiny;
 }
 
 bool SkOpSegment::isTiny(int index) const {
     return fTs[index].fTiny;
 }
 
 // look pair of active edges going away from coincident edge
 // one of them should be the continuation of other
 // if both are active, look to see if they both the connect to another coincident pair
-// if one at least one is a line, then make the pair coincident
+// if at least one is a line, then make the pair coincident
 // if neither is a line, test for coincidence
-bool SkOpSegment::joinCoincidence(SkOpSegment* other, double otherT, int step,
-        bool cancel) {
+bool SkOpSegment::joinCoincidence(SkOpSegment* other, double otherT, int step, bool cancel) {
     int otherTIndex = other->findT(otherT, this);
     int next = other->nextExactSpan(otherTIndex, step);
-    int otherMin = SkTMin(otherTIndex, next);
+    int otherMin = SkMin32(otherTIndex, next);
     int otherWind = other->span(otherMin).fWindValue;
     if (otherWind == 0) {
         return false;
     }
     SkASSERT(next >= 0);
     int tIndex = 0;
     do {
         SkOpSpan* test = &fTs[tIndex];
         SkASSERT(test->fT == 0);
         if (test->fOther == other || test->fOtherT != 1) {
@@ skipping 41 matching lines @@
     SkOpSegment* other = this;
     while ((other = other->nextChase(&index, step, &min, &last))) {
         if (other->done()) {
             return NULL;
         }
         other->markDoneUnary(min);
     }
     return last;
 }
 
-SkOpSpan* SkOpSegment::markAndChaseWinding(const SkOpAngle* angle, const int winding) {
+SkOpSpan* SkOpSegment::markAndChaseWinding(const SkOpAngle* angle, int winding) {
     int index = angle->start();
     int endIndex = angle->end();
     int step = SkSign32(endIndex - index);
     int min = SkMin32(index, endIndex);
     markWinding(min, winding);
     SkOpSpan* last;
     SkOpSegment* other = this;
     while ((other = other->nextChase(&index, step, &min, &last))) {
         if (other->fTs[min].fWindSum != SK_MinS32) {
             SkASSERT(other->fTs[min].fWindSum == winding);
             return NULL;
         }
         other->markWinding(min, winding);
     }
     return last;
 }
 
+SkOpSpan* SkOpSegment::markAndChaseWinding(int index, int endIndex, int winding) {
+    int min = SkMin32(index, endIndex);
+    int step = SkSign32(endIndex - index);
+    markWinding(min, winding);
+    SkOpSpan* last;
+    SkOpSegment* other = this;
+    while ((other = other->nextChase(&index, step, &min, &last))) {
+        if (other->fTs[min].fWindSum != SK_MinS32) {
+            SkASSERT(other->fTs[min].fWindSum == winding || other->fTs[min].fLoop);
+            return NULL;
+        }
+        other->markWinding(min, winding);
+    }
+    return last;
+}
+
 SkOpSpan* SkOpSegment::markAndChaseWinding(int index, int endIndex, int winding, int oppWinding) {
     int min = SkMin32(index, endIndex);
     int step = SkSign32(endIndex - index);
     markWinding(min, winding, oppWinding);
     SkOpSpan* last;
     SkOpSegment* other = this;
     while ((other = other->nextChase(&index, step, &min, &last))) {
         if (other->fTs[min].fWindSum != SK_MinS32) {
             SkASSERT(other->fTs[min].fWindSum == winding || other->fTs[min].fLoop);
             return NULL;
@@ skipping 56 matching lines @@
   //  SkASSERT(!done());
     SkASSERT(winding);
     double referenceT = fTs[index].fT;
     int lesser = index;
     while (--lesser >= 0 && precisely_negative(referenceT - fTs[lesser].fT)) {
         markOneDone(__FUNCTION__, lesser, winding);
     }
     do {
         markOneDone(__FUNCTION__, index, winding);
     } while (++index < fTs.count() && precisely_negative(fTs[index].fT - referenceT));
+    debugValidate();
 }
 
 void SkOpSegment::markDoneBinary(int index) {
     double referenceT = fTs[index].fT;
     int lesser = index;
     while (--lesser >= 0 && precisely_negative(referenceT - fTs[lesser].fT)) {
         markOneDoneBinary(__FUNCTION__, lesser);
     }
     do {
         markOneDoneBinary(__FUNCTION__, index);
     } while (++index < fTs.count() && precisely_negative(fTs[index].fT - referenceT));
+    debugValidate();
 }
 
 void SkOpSegment::markDoneUnary(int index) {
     double referenceT = fTs[index].fT;
     int lesser = index;
     while (--lesser >= 0 && precisely_negative(referenceT - fTs[lesser].fT)) {
         markOneDoneUnary(__FUNCTION__, lesser);
     }
     do {
         markOneDoneUnary(__FUNCTION__, index);
     } while (++index < fTs.count() && precisely_negative(fTs[index].fT - referenceT));
+    debugValidate();
 }
 
 void SkOpSegment::markOneDone(const char* funName, int tIndex, int winding) {
     SkOpSpan* span = markOneWinding(funName, tIndex, winding);
-    if (!span) {
+    if (!span || span->fDone) {
         return;
     }
     span->fDone = true;
     fDoneSpans++;
 }
 
 void SkOpSegment::markOneDoneBinary(const char* funName, int tIndex) {
     SkOpSpan* span = verifyOneWinding(funName, tIndex);
     if (!span) {
         return;
     }
+    SkASSERT(!span->fDone);
     span->fDone = true;
     fDoneSpans++;
 }
 
 void SkOpSegment::markOneDoneUnary(const char* funName, int tIndex) {
     SkOpSpan* span = verifyOneWindingU(funName, tIndex);
     if (!span) {
         return;
     }
+    if (span->fWindSum == SK_MinS32) {
+        SkDebugf("%s uncomputed\n", __FUNCTION__);
+    }
+    SkASSERT(!span->fDone);
     span->fDone = true;
     fDoneSpans++;
 }
 
 SkOpSpan* SkOpSegment::markOneWinding(const char* funName, int tIndex, int winding) {
     SkOpSpan& span = fTs[tIndex];
-    if (span.fDone) {
+    if (span.fDone && !span.fSmall) {
         return NULL;
     }
 #if DEBUG_MARK_DONE
     debugShowNewWinding(funName, span, winding);
 #endif
     SkASSERT(span.fWindSum == SK_MinS32 || span.fWindSum == winding);
     SkASSERT(abs(winding) <= SkPathOpsDebug::gMaxWindSum);
     span.fWindSum = winding;
     return &span;
 }
 
 SkOpSpan* SkOpSegment::markOneWinding(const char* funName, int tIndex, int winding,
                                       int oppWinding) {
     SkOpSpan& span = fTs[tIndex];
     if (span.fDone && !span.fSmall) {
         return NULL;
     }
 #if DEBUG_MARK_DONE
     debugShowNewWinding(funName, span, winding, oppWinding);
 #endif
     SkASSERT(span.fWindSum == SK_MinS32 || span.fWindSum == winding);
     SkASSERT(abs(winding) <= SkPathOpsDebug::gMaxWindSum);
     span.fWindSum = winding;
     SkASSERT(span.fOppSum == SK_MinS32 || span.fOppSum == oppWinding);
     SkASSERT(abs(oppWinding) <= SkPathOpsDebug::gMaxWindSum);
     span.fOppSum = oppWinding;
+    debugValidate();
     return &span;
 }
 
 // from http://stackoverflow.com/questions/1165647/how-to-determine-if-a-list-of-polygon-points-are-in-clockwise-order
 bool SkOpSegment::clockwise(int tStart, int tEnd) const {
     SkASSERT(fVerb != SkPath::kLine_Verb);
     SkPoint edge[4];
     subDivide(tStart, tEnd, edge);
     int points = SkPathOpsVerbToPoints(fVerb);
     double sum = (edge[0].fX - edge[points].fX) * (edge[0].fY + edge[points].fY);
@@ skipping 82 matching lines @@
 #endif
         span->fUnsortableStart = true;
     } else {
         --span;
 #if DEBUG_UNSORTABLE
         debugShowNewWinding(__FUNCTION__, *span, 0);
 #endif
         span->fUnsortableEnd = true;
     }
     if (!span->fUnsortableStart || !span->fUnsortableEnd || span->fDone) {
+        debugValidate();
         return;
     }
     span->fDone = true;
     fDoneSpans++;
+    debugValidate();
 }
 
 void SkOpSegment::markWinding(int index, int winding) {
 //    SkASSERT(!done());
     SkASSERT(winding);
     double referenceT = fTs[index].fT;
     int lesser = index;
     while (--lesser >= 0 && precisely_negative(referenceT - fTs[lesser].fT)) {
         markOneWinding(__FUNCTION__, lesser, winding);
     }
     do {
         markOneWinding(__FUNCTION__, index, winding);
    } while (++index < fTs.count() && precisely_negative(fTs[index].fT - referenceT));
+    debugValidate();
 }
 
 void SkOpSegment::markWinding(int index, int winding, int oppWinding) {
 //    SkASSERT(!done());
     SkASSERT(winding || oppWinding);
     double referenceT = fTs[index].fT;
     int lesser = index;
     while (--lesser >= 0 && precisely_negative(referenceT - fTs[lesser].fT)) {
         markOneWinding(__FUNCTION__, lesser, winding, oppWinding);
     }
     do {
         markOneWinding(__FUNCTION__, index, winding, oppWinding);
    } while (++index < fTs.count() && precisely_negative(fTs[index].fT - referenceT));
+    debugValidate();
 }
 
 void SkOpSegment::matchWindingValue(int tIndex, double t, bool borrowWind) {
     int nextDoorWind = SK_MaxS32;
     int nextOppWind = SK_MaxS32;
+    // prefer exact matches
     if (tIndex > 0) {
         const SkOpSpan& below = fTs[tIndex - 1];
+        if (below.fT == t) {
+            nextDoorWind = below.fWindValue;
+            nextOppWind = below.fOppValue;
+        }
+    }
+    if (nextDoorWind == SK_MaxS32 && tIndex + 1 < fTs.count()) {
+        const SkOpSpan& above = fTs[tIndex + 1];
+        if (above.fT == t) {
+            nextDoorWind = above.fWindValue;
+            nextOppWind = above.fOppValue;
+        }
+    }
+    if (nextDoorWind == SK_MaxS32 && tIndex > 0) {
+        const SkOpSpan& below = fTs[tIndex - 1];
         if (approximately_negative(t - below.fT)) {
             nextDoorWind = below.fWindValue;
             nextOppWind = below.fOppValue;
         }
     }
     if (nextDoorWind == SK_MaxS32 && tIndex + 1 < fTs.count()) {
         const SkOpSpan& above = fTs[tIndex + 1];
         if (approximately_negative(above.fT - t)) {
             nextDoorWind = above.fWindValue;
             nextOppWind = above.fOppValue;
@@ skipping 133 matching lines @@
 }
 
 void SkOpSegment::setUpWindings(int index, int endIndex, int* sumMiWinding,
         int* maxWinding, int* sumWinding) {
     int deltaSum = spanSign(index, endIndex);
     *maxWinding = *sumMiWinding;
     *sumWinding = *sumMiWinding -= deltaSum;
     SkASSERT(abs(*sumWinding) <= SkPathOpsDebug::gMaxWindSum);
 }
 
-// This marks all spans unsortable so that this info is available for early
-// exclusion in find top and others. This could be optimized to only mark
-// adjacent spans that unsortable. However, this makes it difficult to later
-// determine starting points for edge detection in find top and the like.
-bool SkOpSegment::SortAngles(const SkTArray<SkOpAngle, true>& angles,
-                             SkTArray<SkOpAngle*, true>* angleList,
-                             SortAngleKind orderKind) {
-    bool sortable = true;
-    int angleCount = angles.count();
-    int angleIndex;
-    for (angleIndex = 0; angleIndex < angleCount; ++angleIndex) {
-        const SkOpAngle& angle = angles[angleIndex];
-        angleList->push_back(const_cast<SkOpAngle*>(&angle));
+void SkOpSegment::sortAngles() {
+    int spanCount = fTs.count();
+    if (spanCount <= 2) {
+        return;
+    }
+    int index = 0;
+    do {
+        int fromIndex = fTs[index].fFromAngleIndex;
+        int toIndex = fTs[index].fToAngleIndex;
+        if (fromIndex < 0 && toIndex < 0) {
+            index += 1;
+            continue;
+        }
+        SkOpAngle* baseAngle = NULL;
+        if (fromIndex >= 0) {
+            baseAngle = &this->angle(fromIndex);
+            if (inLoop(baseAngle, spanCount, &index)) {
+                continue;
+            }
+        }
 #if DEBUG_ANGLE
-        (*(angleList->end() - 1))->setID(angleIndex);
+        bool wroteAfterHeader = false;
 #endif
-        sortable &= !(angle.unsortable() || (orderKind == kMustBeOrdered_SortAngleKind
-                    && angle.unorderable()));
-    }
-    if (sortable) {
-        SkTQSort<SkOpAngle>(angleList->begin(), angleList->end() - 1);
-        for (angleIndex = 0; angleIndex < angleCount; ++angleIndex) {
-            if (angles[angleIndex].unsortable() || (orderKind == kMustBeOrdered_SortAngleKind
-                        && angles[angleIndex].unorderable())) {
-                sortable = false;
+        if (toIndex >= 0) {
+            SkOpAngle* toAngle = &this->angle(toIndex);
+            if (!baseAngle) {
+                baseAngle = toAngle;
+                if (inLoop(baseAngle, spanCount, &index)) {
+                    continue;
+                }
+            } else {
+                SkDEBUGCODE(int newIndex = index);
+                SkASSERT(!inLoop(baseAngle, spanCount, &newIndex) && newIndex == index);
+#if DEBUG_ANGLE
+                SkDebugf("%s [%d] tStart=%1.9g [%d]\n", __FUNCTION__, debugID(), fTs[index].fT,
+                        index);
+                wroteAfterHeader = true;
+#endif
+                baseAngle->insert(toAngle);
+            }
+        }
+        int nextFrom, nextTo;
+        int firstIndex = index;
+        do {
+            SkOpSpan& span = fTs[index];
+            SkOpSegment* other = span.fOther;
+            SkOpSpan& oSpan = other->fTs[span.fOtherIndex];
+            int otherAngleIndex = oSpan.fFromAngleIndex;
+            if (otherAngleIndex >= 0) {
+#if DEBUG_ANGLE
+                if (!wroteAfterHeader) {
+                    SkDebugf("%s [%d] tStart=%1.9g [%d]\n", __FUNCTION__, debugID(), fTs[index].fT,
+                            index);
+                    wroteAfterHeader = true;
+                }
+#endif
+                baseAngle->insert(&other->angle(otherAngleIndex));
+            }
+            otherAngleIndex = oSpan.fToAngleIndex;
+            if (otherAngleIndex >= 0) {
+#if DEBUG_ANGLE
+                if (!wroteAfterHeader) {
+                    SkDebugf("%s [%d] tStart=%1.9g [%d]\n", __FUNCTION__, debugID(), fTs[index].fT,
+                            index);
+                    wroteAfterHeader = true;
+                }
+#endif
+                baseAngle->insert(&other->angle(otherAngleIndex));
+            }
+            if (++index == spanCount) {
                 break;
             }
+            nextFrom = fTs[index].fFromAngleIndex;
+            nextTo = fTs[index].fToAngleIndex;
+        } while (fromIndex == nextFrom && toIndex == nextTo);
+        if (baseAngle && baseAngle->loopCount() == 1) {
+            index = firstIndex;
+            do {
+                SkOpSpan& span = fTs[index];
+                span.fFromAngleIndex = span.fToAngleIndex = -1;
+                if (++index == spanCount) {
+                    break;
+                }
+                nextFrom = fTs[index].fFromAngleIndex;
+                nextTo = fTs[index].fToAngleIndex;
+            } while (fromIndex == nextFrom && toIndex == nextTo);
+            baseAngle = NULL;
         }
-    }
-    if (!sortable) {
-        for (angleIndex = 0; angleIndex < angleCount; ++angleIndex) {
-            const SkOpAngle& angle = angles[angleIndex];
-            angle.segment()->markUnsortable(angle.start(), angle.end());
-        }
-    }
-    return sortable;
-}
-
-// set segments to unsortable if angle is unsortable, but do not set all angles
-// note that for a simple 4 way crossing, two of the edges may be orderable even though
-// two edges are too short to be orderable.
-// perhaps some classes of unsortable angles should make all shared angles unsortable, but
-// simple lines that have tiny crossings are always sortable on the large ends
-// OPTIMIZATION: check earlier when angles are added to input if any are unsortable
-// may make sense then to mark all segments in angle sweep as unsortableStart/unsortableEnd
-// solely for the purpose of short-circuiting future angle building around this center
-bool SkOpSegment::SortAngles2(const SkTArray<SkOpAngle, true>& angles,
-                              SkTArray<SkOpAngle*, true>* angleList) {
-    int angleCount = angles.count();
-    int angleIndex;
-    for (angleIndex = 0; angleIndex < angleCount; ++angleIndex) {
-        const SkOpAngle& angle = angles[angleIndex];
-        if (angle.unsortable()) {
-            return false;
-        }
-        angleList->push_back(const_cast<SkOpAngle*>(&angle));
-#if DEBUG_ANGLE
-        (*(angleList->end() - 1))->setID(angleIndex);
+#if DEBUG_SORT
+        SkASSERT(!baseAngle || baseAngle->loopCount() > 1);
 #endif
-    }
-    SkTQSort<SkOpAngle>(angleList->begin(), angleList->end() - 1);
-    // at this point angles are sorted but individually may not be orderable
-    // this means that only adjcent orderable segments may transfer winding
-    return true;
+    } while (index < spanCount);
 }
 
 // return true if midpoints were computed
 bool SkOpSegment::subDivide(int start, int end, SkPoint edge[4]) const {
     SkASSERT(start != end);
     edge[0] = fTs[start].fPt;
     int points = SkPathOpsVerbToPoints(fVerb);
     edge[points] = fTs[end].fPt;
     if (fVerb == SkPath::kLine_Verb) {
         return false;
@@ skipping 81 matching lines @@
 }
 
 void SkOpSegment::TrackOutside(SkTArray<SkPoint, true>* outsidePts, const SkPoint& startPt) {
     int outCount = outsidePts->count();
     if (outCount == 0 || startPt != (*outsidePts)[outCount - 1]) {
         outsidePts->push_back(startPt);
     }
 }
 
 void SkOpSegment::undoneSpan(int* start, int* end) {
-    size_t tCount = fTs.count();
-    size_t index;
+    int tCount = fTs.count();
+    int index;
     for (index = 0; index < tCount; ++index) {
         if (!fTs[index].fDone) {
             break;
         }
     }
     SkASSERT(index < tCount - 1);
     *start = index;
     double startT = fTs[index].fT;
     while (approximately_negative(fTs[++index].fT - startT))
         SkASSERT(index < tCount);
@@ skipping 123 matching lines @@
     SkASSERT(span->fWindValue > 0 || span->fOppValue != 0);
     span->fWindValue = 0;
     span->fOppValue = 0;
     if (span->fTiny || span->fSmall) {
         return;
     }
     SkASSERT(!span->fDone);
     span->fDone = true;
     ++fDoneSpans;
 }
-
-#if DEBUG_SWAP_TOP
-bool SkOpSegment::controlsContainedByEnds(int tStart, int tEnd) const {
-    if (fVerb != SkPath::kCubic_Verb) {
-        return false;
-    }
-    SkDCubic dst = SkDCubic::SubDivide(fPts, fTs[tStart].fT, fTs[tEnd].fT);
-    return dst.controlsContainedByEnds();
-}
-#endif
-
-#if DEBUG_CONCIDENT
-// SkASSERT if pair has not already been added
-void SkOpSegment::debugAddTPair(double t, const SkOpSegment& other, double otherT) const {
-    for (int i = 0; i < fTs.count(); ++i) {
-        if (fTs[i].fT == t && fTs[i].fOther == &other && fTs[i].fOtherT == otherT) {
-            return;
-        }
-    }
-    SkASSERT(0);
-}
-#endif
-
-#if DEBUG_CONCIDENT
-void SkOpSegment::debugShowTs(const char* prefix) const {
-    SkDebugf("%s %s id=%d", __FUNCTION__, prefix, fID);
-    int lastWind = -1;
-    int lastOpp = -1;
-    double lastT = -1;
-    int i;
-    for (i = 0; i < fTs.count(); ++i) {
-        bool change = lastT != fTs[i].fT || lastWind != fTs[i].fWindValue
-                || lastOpp != fTs[i].fOppValue;
-        if (change && lastWind >= 0) {
-            SkDebugf(" t=%1.3g %1.9g,%1.9g w=%d o=%d]",
-                    lastT, xyAtT(i - 1).fX, xyAtT(i - 1).fY, lastWind, lastOpp);
-        }
-        if (change) {
-            SkDebugf(" [o=%d", fTs[i].fOther->fID);
-            lastWind = fTs[i].fWindValue;
-            lastOpp = fTs[i].fOppValue;
-            lastT = fTs[i].fT;
-        } else {
-            SkDebugf(",%d", fTs[i].fOther->fID);
-        }
-    }
-    if (i <= 0) {
-        return;
-    }
-    SkDebugf(" t=%1.3g %1.9g,%1.9g w=%d o=%d]",
-            lastT, xyAtT(i - 1).fX, xyAtT(i - 1).fY, lastWind, lastOpp);
-    if (fOperand) {
-        SkDebugf(" operand");
-    }
-    if (done()) {
-        SkDebugf(" done");
-    }
-    SkDebugf("\n");
-}
-#endif
-
-#if DEBUG_ACTIVE_SPANS || DEBUG_ACTIVE_SPANS_FIRST_ONLY
-void SkOpSegment::debugShowActiveSpans() const {
-    debugValidate();
-    if (done()) {
-        return;
-    }
-#if DEBUG_ACTIVE_SPANS_SHORT_FORM
-    int lastId = -1;
-    double lastT = -1;
-#endif
-    for (int i = 0; i < fTs.count(); ++i) {
-        if (fTs[i].fDone) {
-            continue;
-        }
-        SkASSERT(i < fTs.count() - 1);
-#if DEBUG_ACTIVE_SPANS_SHORT_FORM
-        if (lastId == fID && lastT == fTs[i].fT) {
-            continue;
-        }
-        lastId = fID;
-        lastT = fTs[i].fT;
-#endif
-        SkDebugf("%s id=%d", __FUNCTION__, fID);
-        SkDebugf(" (%1.9g,%1.9g", fPts[0].fX, fPts[0].fY);
-        for (int vIndex = 1; vIndex <= SkPathOpsVerbToPoints(fVerb); ++vIndex) {
-            SkDebugf(" %1.9g,%1.9g", fPts[vIndex].fX, fPts[vIndex].fY);
-        }
-        const SkOpSpan* span = &fTs[i];
-        SkDebugf(") t=%1.9g (%1.9g,%1.9g)", span->fT, xAtT(span), yAtT(span));
-        int iEnd = i + 1;
-        while (fTs[iEnd].fT < 1 && approximately_equal(fTs[i].fT, fTs[iEnd].fT)) {
-            ++iEnd;
-        }
-        SkDebugf(" tEnd=%1.9g", fTs[iEnd].fT);
-        const SkOpSegment* other = fTs[i].fOther;
-        SkDebugf(" other=%d otherT=%1.9g otherIndex=%d windSum=",
-                other->fID, fTs[i].fOtherT, fTs[i].fOtherIndex);
-        if (fTs[i].fWindSum == SK_MinS32) {
-            SkDebugf("?");
-        } else {
-            SkDebugf("%d", fTs[i].fWindSum);
-        }
-        SkDebugf(" windValue=%d oppValue=%d\n", fTs[i].fWindValue, fTs[i].fOppValue);
-    }
-}
-#endif
-
-
-#if DEBUG_MARK_DONE || DEBUG_UNSORTABLE
-void SkOpSegment::debugShowNewWinding(const char* fun, const SkOpSpan& span, int winding) {
-    const SkPoint& pt = xyAtT(&span);
-    SkDebugf("%s id=%d", fun, fID);
-    SkDebugf(" (%1.9g,%1.9g", fPts[0].fX, fPts[0].fY);
-    for (int vIndex = 1; vIndex <= SkPathOpsVerbToPoints(fVerb); ++vIndex) {
-        SkDebugf(" %1.9g,%1.9g", fPts[vIndex].fX, fPts[vIndex].fY);
-    }
-    SkASSERT(&span == &span.fOther->fTs[span.fOtherIndex].fOther->
-            fTs[span.fOther->fTs[span.fOtherIndex].fOtherIndex]);
-    SkDebugf(") t=%1.9g [%d] (%1.9g,%1.9g) tEnd=%1.9g newWindSum=%d windSum=",
-            span.fT, span.fOther->fTs[span.fOtherIndex].fOtherIndex, pt.fX, pt.fY,
-            (&span)[1].fT, winding);
-    if (span.fWindSum == SK_MinS32) {
-        SkDebugf("?");
-    } else {
-        SkDebugf("%d", span.fWindSum);
-    }
-    SkDebugf(" windValue=%d\n", span.fWindValue);
-}
-
-void SkOpSegment::debugShowNewWinding(const char* fun, const SkOpSpan& span, int winding,
-                                      int oppWinding) {
-    const SkPoint& pt = xyAtT(&span);
-    SkDebugf("%s id=%d", fun, fID);
-    SkDebugf(" (%1.9g,%1.9g", fPts[0].fX, fPts[0].fY);
-    for (int vIndex = 1; vIndex <= SkPathOpsVerbToPoints(fVerb); ++vIndex) {
-        SkDebugf(" %1.9g,%1.9g", fPts[vIndex].fX, fPts[vIndex].fY);
-    }
-    SkASSERT(&span == &span.fOther->fTs[span.fOtherIndex].fOther->
-            fTs[span.fOther->fTs[span.fOtherIndex].fOtherIndex]);
-    SkDebugf(") t=%1.9g [%d] (%1.9g,%1.9g) tEnd=%1.9g newWindSum=%d newOppSum=%d oppSum=",
-            span.fT, span.fOther->fTs[span.fOtherIndex].fOtherIndex, pt.fX, pt.fY,
-            (&span)[1].fT, winding, oppWinding);
-    if (span.fOppSum == SK_MinS32) {
-        SkDebugf("?");
-    } else {
-        SkDebugf("%d", span.fOppSum);
-    }
-    SkDebugf(" windSum=");
-    if (span.fWindSum == SK_MinS32) {
-        SkDebugf("?");
-    } else {
-        SkDebugf("%d", span.fWindSum);
-    }
-    SkDebugf(" windValue=%d\n", span.fWindValue);
-}
-#endif
-
-#if DEBUG_SORT || DEBUG_SWAP_TOP
-void SkOpSegment::debugShowSort(const char* fun, const SkTArray<SkOpAngle*, true>& angles,
-                                int first, const int contourWinding,
-                                const int oppContourWinding, bool sortable) const {
-    if (--SkPathOpsDebug::gSortCount < 0) {
-        return;
-    }
-    if (!sortable) {
-        if (angles.count() == 0) {
-            return;
-        }
-        if (first < 0) {
-            first = 0;
-        }
-    }
-    SkASSERT(angles[first]->segment() == this);
-    SkASSERT(!sortable || angles.count() > 1);
-    int lastSum = contourWinding;
-    int oppLastSum = oppContourWinding;
-    const SkOpAngle* firstAngle = angles[first];
-    int windSum = lastSum - spanSign(firstAngle);
-    int oppoSign = oppSign(firstAngle);
-    int oppWindSum = oppLastSum - oppoSign;
-    #define WIND_AS_STRING(x) char x##Str[12]; \
-            if (!SkPathOpsDebug::ValidWind(x)) strcpy(x##Str, "?"); \
-            else SK_SNPRINTF(x##Str, sizeof(x##Str), "%d", x)
-    WIND_AS_STRING(contourWinding);
-    WIND_AS_STRING(oppContourWinding);
-    SkDebugf("%s %s contourWinding=%s oppContourWinding=%s sign=%d\n", fun, __FUNCTION__,
-            contourWindingStr, oppContourWindingStr, spanSign(angles[first]));
-    int index = first;
-    bool firstTime = true;
-    do {
-        const SkOpAngle& angle = *angles[index];
-        const SkOpSegment& segment = *angle.segment();
-        int start = angle.start();
-        int end = angle.end();
-        const SkOpSpan& sSpan = segment.fTs[start];
-        const SkOpSpan& eSpan = segment.fTs[end];
-        const SkOpSpan& mSpan = segment.fTs[SkMin32(start, end)];
-        bool opp = segment.fOperand ^ fOperand;
-        if (!firstTime) {
-            oppoSign = segment.oppSign(&angle);
-            if (opp) {
-                oppLastSum = oppWindSum;
-                oppWindSum -= segment.spanSign(&angle);
-                if (oppoSign) {
-                    lastSum = windSum;
-                    windSum -= oppoSign;
-                }
-            } else {
-                lastSum = windSum;
-                windSum -= segment.spanSign(&angle);
-                if (oppoSign) {
-                    oppLastSum = oppWindSum;
-                    oppWindSum -= oppoSign;
-                }
-            }
-        }
-        SkDebugf("%s [%d] %s", __FUNCTION__, index,
-                angle.unsortable() ? "*** UNSORTABLE *** " : "");
-    #if DEBUG_SORT_COMPACT
-        SkDebugf("id=%d %s start=%d (%1.9g,%1.9g) end=%d (%1.9g,%1.9g)",
-                segment.fID, kLVerbStr[SkPathOpsVerbToPoints(segment.fVerb)],
-                start, segment.xAtT(&sSpan), segment.yAtT(&sSpan), end,
-                segment.xAtT(&eSpan), segment.yAtT(&eSpan));
-    #else
-        switch (segment.fVerb) {
-            case SkPath::kLine_Verb:
-                SkDebugf(LINE_DEBUG_STR, LINE_DEBUG_DATA(segment.fPts));
-                break;
-            case SkPath::kQuad_Verb:
-                SkDebugf(QUAD_DEBUG_STR, QUAD_DEBUG_DATA(segment.fPts));
-                break;
-            case SkPath::kCubic_Verb:
-                SkDebugf(CUBIC_DEBUG_STR, CUBIC_DEBUG_DATA(segment.fPts));
-                break;
-            default:
-                SkASSERT(0);
-        }
-        SkDebugf(" tStart=%1.9g tEnd=%1.9g", sSpan.fT, eSpan.fT);
-    #endif
-        SkDebugf(" sign=%d windValue=%d windSum=", angle.sign(), mSpan.fWindValue);
-        SkPathOpsDebug::WindingPrintf(mSpan.fWindSum);
-        int last, wind;
-        if (opp) {
-            last = oppLastSum;
-            wind = oppWindSum;
-        } else {
-            last = lastSum;
-            wind = windSum;
-        }
-        bool useInner = SkPathOpsDebug::ValidWind(last) && SkPathOpsDebug::ValidWind(wind)
-                && UseInnerWinding(last, wind);
-        WIND_AS_STRING(last);
-        WIND_AS_STRING(wind);
-        WIND_AS_STRING(lastSum);
-        WIND_AS_STRING(oppLastSum);
-        WIND_AS_STRING(windSum);
-        WIND_AS_STRING(oppWindSum);
-        #undef WIND_AS_STRING
-        if (!oppoSign) {
-            SkDebugf(" %s->%s (max=%s)", lastStr, windStr, useInner ? windStr : lastStr);
-        } else {
-            SkDebugf(" %s->%s (%s->%s)", lastStr, windStr, opp ? lastSumStr : oppLastSumStr,
-                    opp ? windSumStr : oppWindSumStr);
-        }
-        SkDebugf(" done=%d unord=%d small=%d tiny=%d opp=%d\n",
-                mSpan.fDone, angle.unorderable(), mSpan.fSmall, mSpan.fTiny, opp);
-        ++index;
-        if (index == angles.count()) {
-            index = 0;
-        }
-        if (firstTime) {
-            firstTime = false;
-        }
-    } while (index != first);
-}
-
-void SkOpSegment::debugShowSort(const char* fun, const SkTArray<SkOpAngle*, true>& angles,
-                                int first, bool sortable) {
-    if (!sortable) {
-        if (angles.count() == 0) {
-            return;
-        }
-        if (first < 0) {
-            first = 0;
-        }
-    }
-    const SkOpAngle* firstAngle = angles[first];
-    const SkOpSegment* segment = firstAngle->segment();
-    int winding = segment->updateWinding(firstAngle);
-    int oppWinding = segment->updateOppWinding(firstAngle);
-    debugShowSort(fun, angles, first, winding, oppWinding, sortable);
-}
-
-#endif
-
-#if DEBUG_SHOW_WINDING
-int SkOpSegment::debugShowWindingValues(int slotCount, int ofInterest) const {
-    if (!(1 << fID & ofInterest)) {
-        return 0;
-    }
-    int sum = 0;
-    SkTArray<char, true> slots(slotCount * 2);
-    memset(slots.begin(), ' ', slotCount * 2);
-    for (int i = 0; i < fTs.count(); ++i) {
-   //     if (!(1 << fTs[i].fOther->fID & ofInterest)) {
-   //         continue;
-   //     }
-        sum += fTs[i].fWindValue;
-        slots[fTs[i].fOther->fID - 1] = as_digit(fTs[i].fWindValue);
-        sum += fTs[i].fOppValue;
-        slots[slotCount + fTs[i].fOther->fID - 1] = as_digit(fTs[i].fOppValue);
-    }
-    SkDebugf("%s id=%2d %.*s | %.*s\n", __FUNCTION__, fID, slotCount, slots.begin(), slotCount,
-            slots.begin() + slotCount);
-    return sum;
-}
-#endif
-
-void SkOpSegment::debugValidate() const {
-#if DEBUG_VALIDATE
-    int count = fTs.count();
-    SkASSERT(count >= 2);
-    SkASSERT(fTs[0].fT == 0);
-    SkASSERT(fTs[count - 1].fT == 1);
-    int done = 0;
-    double t = -1;
-    for (int i = 0; i < count; ++i) {
-        const SkOpSpan& span = fTs[i];
-        SkASSERT(t <= span.fT);
-        t = span.fT;
-        int otherIndex = span.fOtherIndex;
-        const SkOpSegment* other = span.fOther;
-        const SkOpSpan& otherSpan = other->fTs[otherIndex];
-        SkASSERT(otherSpan.fPt == span.fPt);
-        SkASSERT(otherSpan.fOtherT == t);
-        SkASSERT(&fTs[i] == &otherSpan.fOther->fTs[otherSpan.fOtherIndex]);
-        done += span.fDone;
-    }
-    SkASSERT(done == fDoneSpans);
-#endif
-}
-
-#ifdef SK_DEBUG
-void SkOpSegment::dumpPts() const {
-    int last = SkPathOpsVerbToPoints(fVerb);
-    SkDebugf("{{");
-    int index = 0;
-    do {
-        SkDPoint::dump(fPts[index]);
-        SkDebugf(", ");
-    } while (++index < last);
-    SkDPoint::dump(fPts[index]);
-    SkDebugf("}}\n");
-}
-
-void SkOpSegment::dumpDPts() const {
-    int count = SkPathOpsVerbToPoints(fVerb);
-    SkDebugf("{{");
-    int index = 0;
-    do {
-        SkDPoint dPt = {fPts[index].fX, fPts[index].fY};
-        dPt.dump();
-        if (index != count) {
-            SkDebugf(", ");
-        }
-    } while (++index <= count);
-    SkDebugf("}}\n");
-}
-
-void SkOpSegment::dumpSpans() const {
-    int count = this->count();
-    for (int index = 0; index < count; ++index) {
-        const SkOpSpan& span = this->span(index);
-        SkDebugf("[%d] ", index);
-        span.dump();
-    }
-}
-#endif